Lie Theory and Special Functions
Mathematics in Science and Engineering A Series
of Monographs and
Textbooks
Edited by RICHARD BELLMAN, University of Southern California
1. TRACY Y. THOMAS. Concepts from Tensor Analysis and Differential Geometry. Second Edition. 1965
Y. THOMAS. Plastic Flow and Fracture in Solids. 1961 2. TRACY ARIS. The Optimal Design of Chemical Reactors: A Study in Dynamic 3. RUTHERFORD Programming. 1961
4. JOSEPHLASALLEand SOLOMON LEFSCHETZ. Stability by Liapunov’s Direct Method with Applications. 1961
(ed.) . Optimization Techniques: With Applications to Aero5. GEORGELEITMANN
6. 7. 8. 9.
space Systems. 1962 RICHARD BELLMAN and KENNETH L. COOKE.Differential-Difference Equations. 1963
FRANK A. HAIGHT.Mathematical Theories of Traffic Flow. 1963 F. V. ATKINSON.Discrete and Continuous Boundary Problems. 1964 A. JEFFREYand T. TANIUTI. Non-Linear Wave Propagation: With Applications to Physics and Magnetohydrodynamics. 1964
10. JULIUST. Tou. Optimum Design of Digital Control Systems. 1963 11. HARLEYFLANDERS. Differential Forms: With Applications to the Physical Sciences. 1963 M. ROBERTS. Dynamic Programming in Chemical Engineering and Process 12. SANFORD Control. 1964
13. 14. 15. 16. 17.
SOLOMON LEFSCHETZ. Stability of Nonlinear Control Systems. 1965 DIMITRISN. CHORAFAS. Systems and Simulation. 1965 A. A. PERVOZVANSKII. Random Processes in Nonlinear Control Systems. 1965 111, Methods of Matrix Algebra. 1965 MARSHALL C. PEASE,
V. E. BENES.Mathematical Theory of Connecting Networks and Telephone Traffic. 1965 18. WILLIAMF. AMES. Nonlinear Partial Differential Equations in Engineering. 1965
19. 20. 21. 22. 23. 24. 25. 26. 27.
J. ACZEL.Lectures on Functional Equations and Their Applications. 1966 R. E. MURPHY.Adaptive Processes in Economic Systems. 1965 S. E. DREYFUS.Dynamic Programming and the Calculus of Variations. 1965 A. A. FEL’DBAUM. Optimal Control Systems. 1965 A. HALANAY. Differential Equations : Stability, Oscillations, Time Lags. 1966 M. NAMIKOCUZTOWI.Time-Lag Control Systems. 1966 DAVIDSWORDER. Optimal Adaptive Control Systems. 1966
MILTONASH. Optimal Shutdown Control of Nuclear Reactors. 1966 DIMITRISN. CHORAFAS. Control System Functions and Programming Approaches ( I n Two Volumes). 1966 28. N. P. ERUGIN.Linear Systems of Ordinary Differential Equations. 1966 MARCUS.Algebraic Linguistics; Analytical Models. 1967 29. SOLOMON Stability of Motion. 1966 30. A. M. LIAPUNOV.
31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42.
43. 44. 45.
.
GEORGE LEITMANN (ed.) Topics i n Optimization. 1967 MASANAO AOKI. Optimization of Stochastic Systems. 1967 J. KUSHNER. Stochastic Stability and Control. 1967 HAROLD MINORUURABE. Nonlinear Autonomous Oscillations. 1967 F. CALOGERO. Variable Phase Approach to Potential Scattering. 1967 A. KAUFMANN. Graphs, Dynamic Programming, and Finite Games. 1967 A. KAUFMANN and R. CRUON.Dynamic Programming: Sequential Scientific Management. 1967 J. H. AHLBERG, E. N. NILSON,and J. L. WALSH.The Theory of Splines and Their Applications. 1967 Y. SAWARACI, Y. SUNAHARA, and T. NAKAMIZO. Statistical Decision Theory in Adaptive Control Systems. 1967 RICHARD BELLMAN. Introduction to the Mathematical Theory of Control Processes Volume I. 1967 (Volumes I1 and I11 in preparation) E. STANLEY LEE. Quasilinearization and Invariant Imbedding. 1968 WILLIAM AMES.Nonlinear Ordinary Differential Equations in Transport Processes. 1968 WILLARD MILLER,JR. Lie Theory and Special Functions. 1968 PAULB. BAILEY,LAWRENCE F. SHAMPINE, and PAUL E. WALTMAN. Nonlinear Two Point Boundary Value Problems. 1968 P. PETROV. Variational Methods in Optimum Control Theory. 1968
In preparation
A. KAUFMANN and R. FAURE.Introduction to Operations Research N. K. JAISWAL. Priority Queues ROBERT HERMANN. Differential Geometry and the Calculus of Variations 0. A. LADYZHENSKAYA and N. N. URAL’TSEVA. Linear and Quasilinear Elliptic Equations Comparison and Oscillation Theory of Differential Equations C. A. SWANSON. ROBERT P. GILBERT. Function Theoretic Methods in the Theory of Partial Differential Equations YUDELL LUKE.The Special Functions and Their Approximations (In Two Volumes) HUKUKANE NIKAIDO.Convex Structures and Economic Theory
This page intentionally left blank
Lie Theory and Special Functions
Willard Miller, Jr. DEPARTMENT OF MATHEMATICS UNIVERSITY OF MINNESOTA MINNEAPOLIS, MINNESOTA
Academic Press
New York and London
1968
COPYRIGHT 0 1968,
BY ACADEMIC PRESS ALL RIGHTS RESERVED.
INC.
NO PART OF THIS BOOK MAY BE REPRODUCED I N ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.
ACADEMIC PRESS INC. 111 Fifth Avenue, New York, New York 10003
United Kingdom Edition published by ACADEMIC PRESS INC. (LONDON) LTD. Berkeley Square House, London W.l
LIBRARY OF CONGRESS CATALOG CARDNUMBER: 68-18677
PRINTED I N THE UNITED STATES OF AMERICA
In Memory of Professor Bernard Friedman
This page intentionally left blank
Preface
This monograph is the result of an attempt to understand the role played by special function theory in the formalism of mathematical physics. I t demonstrates explicitly that special functions which arise in the study of mathematical models of physical phenomena and the identities which these functions obey are in many cases dictated by symmetry groups admitted by the models. I n particular it will be shown that the factorization method, a powerful tool for computing eigenvalues and recurrence relations for solutions of second order ordinary differential equations (Infeld and Hull [l]), is equivalent to the representation theory of four local Lie groups. A detailed study of these four groups and their Lie algebras leads to a unified treatment of a significant proportion of special function theory, especially that part of the theory which is most useful in mathematical physics. Most of the identities for special functions derived in this book are known in one form or another. Our principal aim is not to derive new results but rather to provide insight into the structure of special function theory. Thus, all of the identities obtained here will be given an explicit group-theoretic interpretation instead of being considered merely as the result of some formal manipulation of infinite series. T h e primary tools needed to deduce our results are multiplier representations of local Lie groups and representations of Lie algebras by generalized Lie derivatives. These concepts are introduced in Chapter 1 along with a brief survey of classical Lie theory. I n Chapter 2 we state our main theme: Special functions occur as matrix elements and basis vectors corresponding to multiplier representations of local Lie groups. Chapters 3-6 are devoted to the explicit analysis of the special function ix
X
PREFACE
theory of the complex local Lie groups with four-dimensional Lie . algebras g(0, O), g(0, l), %(I, 0) and six-dimensional Lie algebra F6 Many fundamental properties of hypergeometric, confluent hypergeometric, and Bessel functions are obtained from this analysis. Furthermore, in Chapter 7 it is shown that the representation theory of these four Lie groups is completely equivalent to the factorization method of Infeld and Hull. I n Chapter 8 we determine the scope of our analysis by constructing the rudiments of a classification theory of generalized Lie derivatives. T hi s classification theory enables us to decide in what sense the results of Chapters 3-6 are complete. Chapter 8 is more difficult than the rest of the book since it presupposes a knowledge of some rather deep results in local Lie theory and an acquaintance with the cohomology theory of Lie algebras. Hence, it may be omitted on a first reading. Finally, in Chapter 9 we apply some of the results of the classification theory to obtain identities for special functions which are not related to the factorization method. It should be noted that we will be primarily concerned with the representation theory of local Lie groups, a subject which was developed in the nineteenth century. T h e more recent and sophisticated theory of global Lie groups is by itself too narrow to obtain many fundamental identities for special functions. Also, unitary representations of Lie groups will occur only as special cases of our results; they will not be our primary concern. T h e scope of this book is modest: we study no Lie algebras with , dimension greater than 6. Furthermore, in the six-dimensional case, F6 we do not give complete results. (The so-called addition theorems of Gegenbauer type for Bessel functions would be obtained from such an analysis.) However, it should be clear to the reader that our methods can be generalized to higher dimensional Lie algebras. We will almost exclusively be concerned with the derivation of recursion relations and addition theorems. T h e manifold applications of group theory in the derivation of orthogonality relations and integral transforms of special functions will rarely be considered. For these applications see the encyclopedic work of Vilenkin [3]. T h e overlap in subject matter between that book and this one is relatively small, except in the study of unitary representations of real Lie groups. This monograph has been strongly influenced by the work of Bargmann [ 1 4 ] , Infeld and Hull [I], Schwinger [I], Weisner [l-31, and Wigner [l, 21. T h e author’s primary contribution has been to weave these papers into a coherent theory relating Lie groups and special functions. A knowledge of the basic elements of complex analysis, functional
PREFACE
xi
analysis, local Lie theory, and the theory of special functions is desirable for a proper understanding of the work presented here. This would correspond roughly with the background of an advanced graduate student in theoretical physics. I wish to dedicate this book to the memory of Professor Bernard Friedman, in the hope that it will in a small way help to bridge the gap between the pure and applied sciences, a cause for which he was such an eloquent advocate.
January 1968
WILLARD MILLER,JR.
This page intentionally left blank
Contents PREFACE .
...............................
ix
CHAPTER 1
RCum6 of l i e Theory 1-1 1-2 1-3 1-4
Local Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . Local Transformation Groups . . . . . . . . . . . . . . . Examples of Local Transformation Groups . . . . . . . . .
.. . .
1 5 12 19
CHAPTER 2
Representations and Realizations of Lie Algebras 2-1 Representations of Lie Algebras . . . . . . 2-2 Realizations of Representations . . . . . . . 2-3 Representations of L ( 0 , ) . . . . . . . . . 2-4 The Angular Momentum Operators . . . . 2-5 The Lie Algebras s(a.b) . . . . . . . . . 2-6 Representations of S(a. b) . . . . . . . . . 2-7 Realizations of S(a. b) in Two Variables . . . 2-8 Realizations of '3(a. b) in One Variable . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . .
.......... . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
25 27 31 33 36 38 45 49
CHAPTER 3
l i e Theory and Bessel Functions 3-1 The Representations Q(w. m.) . . . . . . 3-2 Recursion Relations for the Matrix Elements 3-3 Realizations of Q(w. m. ) in Two Variables . 3-4 Weisner's Method for Bessel Functions . . 3-5 The Real Euclidean Group E. . . . . . . 3-6 Unitary Representations of Lie Groups . . . . . . . . 3-7 Induced Representations of xiii
. . . . . . . . . . .
...........
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
52 57 59 63 65 67 70
CONTENTS
xiv
3-8 The Unitary Representations ( p ) of E. 3-9 The Matrix Elements of ( p ) . . . . 3-10 The Infinitesimal Operators of ( p ) .
..............
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
71 75 76
CHAPTER 4
Lie Theory and Confluent H ypergoemetric Functions 4-1 The Representation R(w. m., p) . . . . . . . . 4-2 4-3 4-4 4-5 4-6 4-7 4-8
4-9 4-10 4-11 4-12 4-1 3 4-14 4-15 4-16 4-17 4-18 4-19 4-20 4-21
........
The Representation to. . . . . . . . . . . . . . . . . . . . TheRepresentation bW. . . . . . . . . . . . . . . . . . . . Differential Equations for the Matrix Elements . . . . . . . . . . The Representation @ ............... A Realization of R(w. m. p ) by Type D Operators . . . . . . . . A Realization of to.,,by Type D Operators . . . . . . . . . . . Transformations of Type C' Operators . . . . . . . . . . . . . Type C' Realizations of R(w. m. p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Type C' Realizations of The Group S4 . . . . . . . . . . . . . . . . . . . . . . . . Induced Representations of 9. . . . . . . . . . . . . . . . . . The Hilbert Space S . . . . . . . . . . . . . . . . . . . . . The Unitary Representation (A. I) . . . . . . . . . . . . . . . . The Matrix Elements of (A. I) . . . . . . . . . . . . . . . . . The Unitary Representations (A. -I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Tensor Products (A. I) Q (A'. l') The Representations (A. I) Q (A'. 4') .............. The Representations (A. I) Q (A'. 4). . . . . . . . . . . . . . The Representations ( p ) Q (A. I) . . . . . . . . . . . . . . . . A Contraction of Y(O.1) . . . . . . . . . . . . . . . . . . .
tolSpl t%.,,
.
.
79 85 89 91 94 101 104 106 109
111 113 114 117 119 122 125 127 131 136 146 150
CHAPTER 5
Lie Theory and Hypcrgoemetric Functions 5-1 The Representation P ( u . m.,) . . . .
tu
.............
5-2 The Representation .................... 5-3 The Representation ,, . . . . . . . . . . . . . . . . . . . . 5-4 The Representation D(2u) . . . . . . . . . . . . . . . . . . . 5-5 The Tensor Product D(2u) Q D(2u) . . . . . . . . . . . . . . . 5-6 The Tensor Product Q . . . . . . . . . . . . . . . . . . 5-7 Differential Relations for the Matrix Elements . . . . . . . . . . 5-8 Type B Realizations of D(u. m.) . . . . . . . . . . . . . . . . 5-9 Type B Realizations of . . . . . . . . . . . . . . . . . . . . 5-10 Weisner's Method for Type B Operators . . . . . . . . . . . . 5-11 Type A Realizations of D(u. m.,) . . . . . . . . . . . . . . . . 5- 12 Type A Realizations of . . . . . . . . . . . . . . . . . . . . 5-13 Type A Realizations of . . . . . . . . . . . . . . . . . . . . 5-14 Type A Realizations of D(2u) . . . . . . . . . . . . . . . . . 5-15 Weisner's Method for Type A Operators . . . . . . . . . . . . . 5-16 The Group SU(2) . . . . . . . . . . . . . . . . . . . . . . . 5-17 The Group G. . . . . . . . . . . . . . . . . . . . . . . . . 5-18 Unitary Representations of G, . . . . . . . . . . . . . . . . . 5-19 Contractions of 9(1. 0 ) . . . . . . . . . . . . . . . . . . . .
4
tu t t
t -1
155 161 165 167 170 176 180 183 187 190 194 200 205 206 210 214 221 226 238
CONTENTS
xv
CHAPTER 6
Special Functions Related to the Euclidean Group in I S p a c e Representations of .Te. . . . . . . . . . Type E Operators . . . . . . . . . . . Type F Operators . . . . . . . . . . . The Euclidean Group Es . . . . . . . The Matrix Elements of ( w . s) . . . . .
6-1 6-2 6-3 6-4 6-5
........... ........... ...........
............ . . . . . . . . . . . .
243 249 253 255 262
CHAPTER 7
The Factorization Method
7- 1 Recurrence Relations . . 7-2 The Factorization Types .
................... ...................
268 272
CHAPTER 8
Generalized Lie Derivatives
8-1 Generalized Derivations . . . . . . . . . . 8-2 Cohomology Classes of Realizations . . . . 8-3 Computation of the Cohomology Classes . 8-4 Tables of Cohomology Classes . . . . . .
. . . .
. . . .
........ ......... ......... . . . . . . . . .
278 281 289 293
The Lie Algebra X6 . . . . . . . . . . . . . . . . . . . . . The Lie Group K I . . . . . . . . . . . . . . . . . . . . . . Representations of X5 . . . . . . . . . . . . . . . . . . . . The Representation R'(w. m,. p ) . . . . . . . . . . . . . . . . The Representation . . . . . . . . . . . . . . . . . . . The Lie Algebra Z'J., . . . . . . . . . . . . . . . . . . . . . . The Lie Group G,.. . . . . . . . . . . . . . . . . . . . . . Representations of 'ZJn.@ . . . . . . . . . . . . . . . . . . . . Recursion Relations for the Matrix Elements . . . . . . . . . . . Realizations in Two Variables . . . . . . . . . . . . . . . . . Generating Functions for Generalized Bessel Functions . . . . . .
299 300 301 302 308 312 313 314 316 318 320
CHAPTER 9
Some Generalizations 9-1 9-2 9-3 9-4 9-5 9-6 9-7 9-8 9-9 9-10 9-11
Appendix I 2 3 4 5
t;.
The Gamma Function . . . . . . . . . . The Hypergeometric Function . . . . . . The Confluent Hypergeometric Function . Parabolic Cylinder Functions . . . . . . Bessel Functions . . . . . . . . . . . . .
........... . . . . . . . . . . . . . . . . . . . . . . . . ............ ...........
324 325 327 328 329
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
330
.............................
336
INDEX
This page intentionally left blank
Lie Theory ond Special Functions
This page intentionally left blank
CHAPTER
1 R6sumi5 of Lie Theory
This chapter contains the fundamental definitions and results from the theory of local Lie groups that will be needed in the remainder of the book. The basic theorems are presented largely without proofs, since the material is treated in standard works on Lie theory (Lie [I], Cohn [I], Pontrjagin [l], Guggenheimer [l]). Although a good understanding of these theorems is desirable for a proper appreciation of group-theoretic methods in special function theory, it is not essential, as we will be primarily concerned with specific examples for which we can directly verify the validity of our group-theoretic manipulations. The treatment of Lie theory given here is based on that of Guggenheimer [l], which is a reformulation of Lie’s original approach. This treatment is characterized by its dependence on local coordinates. T h e study of special functions requires detailed computations in explicitly given coordinate systems, so modern coordinatefree methods in Lie theory are not particularly useful. The examples of local Lie groups and local Lie transformation groups presented in Sections 1-2 and 1-4 are important since they will recur frequently throughout the book.
1-1
Local Lie Groups
I t is assumed that the reader is familiar with the concept of a (global) Lie group: A Lie group is both an abstract group and an analytic manifold such that the operations of group multiplication and group inversion are analytic with respect to the manifold structure (Hamermesh [I],
1. RESUME OF LIE THEORY
2
Helgason [l], Pontrjagin [l], Eisenhart [l]). I n much of this book, however, we shall be concerned with groups which are defined and analytic only in some neighborhood of the group identity element, that is, with local Lie groups. Let @, be the space of complex n-tuples g = (g, ,g,,...,g,) where gi E @ for i = 1, 2,..., n, and define the origin e of @% by e = ( 0 , O ,..., 0). Suppose V is an open set in @n containing e. Definition A complex n-dimensional local Lie group G in the neighborhood V C gn is determined by a function cp: x @n + such that
cn
(1)
cp(g,h) E @"
for all
en
g, h E V.
(2) V(g, h) is analytic in each of its 2n arguments. (3) Ifcp(g, h) E V , v(h, k) E V , then cp(cp(g,h), k) = cp(g, cp(h, k)). (4) cp(e, g) = g, cp(g, e) = g for all g E V . I n particular, a neighborhood of the identity element of a global Lie group is a local Lie group. It follows from (4) and the implicit function theorem that if G is a local Lie group there exists a neighborhood V' of e such that for every g E V' there is a unique element denoted g-l in V with the property cp(g, g-l) = V(g-', g) = e . T h e mapping p(g) = g-l is analytic in the n arguments of g. T o simplify notation, we shall often write cp(g, h) = gh. I n terms of this definition of group multiplication, property ( 3 ) is the associative law (gh)k = g(hk) and property (.4) defines e as the group identity element. Furthermore, if g is in a small enough neighborhood of e then 8-l E V where g-l is the multiplicative inverse of g. Let G be a local Lie group and let t + g(t) = (gl(t),...,g,(t)), t E @, be an analytic mapping of a neighborhood of 0 E @ into V such that g(0) = e. We can consider such a mapping to be a complex analytic curve in G passing through e. T h e tangent vector to g(t) at e is the vector
Every vector CY E gn is the tangent vector at e for some analytic curve. I n particular, the curve at = (OLlt, OLZt,..., ant)
has the tangent vector
CY
=
(01,
,..., an) at e .
(1.2)
1-1. LOCAL LIE GROUPS
3
If g ( t ) , h(t) are analytic curves in G with g(0) = h(0) = e and tangent vectors 01, 8 at e, respectively, then the analytic curve g(t)h(t)has tangent /3 at e. Here, the plus sign (+) refers to the usual vector vector 01 addition in Cn. With g ( t ) , h(t) as given above, define the commutator [ 0 1 , / 3 ] of 01 and /3 to be the tangent vector at e of the analytic curve
+
k(t) = g(T)h(T) g-'(T) h-'(T),
t
=T
(1.3)
~ .
Thus,
T h e commutator has the properties:
(1) [a, 81 = -[851,
(2)
[al%
+
a2a2 >
011,
PI
a51,
=a 1 C . 1
8 e Cn,
, 81 + a2[O12
7
PI,
+
a1 011
3
a2 E
C51,
8 E en,
, 01.2,
(3) "a, 8151,rl "P51, rl51,a1 = "a, rl, 61. T h e Lie algebra L( G) of the local Lie group G is the set of all tangent vectors at e equipped with the operations of vector addition and Lie product. Definition A complex abstract Lie algebra 9 is a complex vector space together with a multiplication [a, /3] E 9 defined for all a, 8 E 9 such that
81 = -[/351, 4, (2) [%% + aza2 81 = %[a1 81 + 4%PI,
(1)
[a,
51,
(3)
"%
8151,rl
+ "8, rl, a3
7
9
= "%
71, 81.
a, 011
a2 E @, , 012 8 E 9, 7
7
Clearly, L(G) is a complex abstract Lie algebra. (Note: Unless otherwise stated all Lie algebras are assumed to be finite-dimensional. See Section 2-1 for a definition of this concept.) Definition An analytic homomorphism of a local Lie group G into a local Lie group G' is a map p: G + G', where p ( g ) is defined for g in a suitably small neighborhood W of e, such that
CLW) = Ad Ah),
g, h, gh E
w,
(1.5)
and p is an analytic function of the coordinates of G. T h e group multiplication on the right-hand side of (1 S ) takes place in G'.
1. Rl%UME OF LIE THEORY
4
It follows from (1.5) that p(e) = e', where e' is the identity element of G', and p(g-') = p(g)-l for g in a suitably small neighborhood of e in G. A homomorphism which maps G one-to-one onto a neighborhood of e' in G' is called an isomorphism. An isomorphism of G onto G is a n automorphism. Note that an element h of G defines an automorphism &)
= hgh-l,
g E G-
(1.6)
Automorphisms of the form (1.6) are called inner automorphisms. Let 9,9'be complex abstract Lie algebras with operations +, [., .] and [., .]', respectively.
+',
Definition A homomorphism from 3 to 9'is a map such that
(1)
(2)
T(aO1 $-
PI)
r([a,
bp) = aT(a) f '
bT(@),
a, 01,
4311'.
E
T:
9 -+ 3'
@,
fi E 9,
= [7(a>,
A Lie algebra homomorphism which is a one-to-one map of 3 onto 9' is called an isomorphism. An isomorphism of 9 onto 3 is an automorphism. Theorem 1.1 An analytic homomorphism p : G -+G' of the local Lie groups G, G' induces a Lie algebra homomorphism p * : L(G)-tL(G') defined by
where g ( t ) is an analytic curve in G with tangent vector 01 at e. T h e p* is an isomorphism (automorphism) if and only if p is an isomorphism (automorphism). We can now state the basic Theorem 1.2 (Converse of Lie's Third Fundamental Theorem) A complex abstract Lie algebra is the Lie algebra of some local Lie
group. Two local Lie groups with isomorphic Lie algebras are isomorphic. Thus, up to isomorphism there is a one-to-one correspondence between local Lie groups and Lie algebras. Let G be a local Lie group and let g, h E V . Then
gh
= Cpk,
h)
=
( d g ,h),...,vn.n(g,h))
1-2. EXAMPLES
5
is an analytic function of g and h. This function can be expanded in a Taylor series in the components of g about g = e. Thus,
+ &%Fij(h) + ... n
d g , h ) = hi
( 1-71
3=1
where g= e
Let g ( t ) = (gl(t),...,g,( t ) ) be the integral curve of the system of differential equations
with the initial condition g(0) = e. T h e function g ( t ) is uniquely determined by a E L(G) and is defined for suitably small values of I t I. We write g ( t ) = exp at. Theorem 1.3 T h e integral curve exp at is a 1-parameter subgroup of G such that exp a(tl t,) = exp atl exp at, for suitably small values of I t , 1, 1 t, I. T h e tangent vector of exp at at e is 01 = ( a , ,..., a,). Since the elements of L(G) are n-tuples, we can consider L(G) to be induces a topology in L(G). defined in @. T h e usual topology of
+
Theorem 1.4 T h e mapping a-ex p a is an analytic diffeomorphism of a neighborhood of 0 EL(G) onto a neighborhood of e in G. (Here, 0 is the additive identity ofL(G).) Th a t is, the mapping defines a local one-to-one analytic coordinate transformation in U p to now we have been concerned with the theory of complex local Lie groups. However, there is an analogous theory of real local Lie groups. The relevant definitions and theorems for real groups can easily be obtained from the foregoing analysis by replacing the word “complex” and the symbol C‘ (complex field) by the word “real” and the symbol R (real field).
en.
1-2
Examples
T h e 2 x 2 complex general linear group GL(2) is the set of all 2 by 2 nonsingular matrices g=
(f
i),
a , b, c, d € Q, ad
-
6c # 0,
(1.9)
1. RESUME
6
OF LIE THEORY
where the group operation is matrix multiplication (Boerner [l], Chapter 2). (We will sometimes use the notation G(l, 0) for GL(2).) Clearly, the identity element of GL(2)is the matrix
I n a neighborhood of e in GL(2) we can introduce coordinates for the group element g by setting g
= (g, g, 7
1
,
g3 g4)
= ( a - 1, b, c, d - 1).
(1.10)
With this coordinate system it is easy to verify that GL(2)is a 4-dimensional complex local Lie group. These coordinates are valid only for g in a suitably small neighborhood of e ; they cannot be extended over all of @4. Any local Lie group obtained from (1.9) and (1.10) by introducing an analytic invertible change of local coordinates is locally isomorphic to the group constructed here. Thus, it makes sense to speak about the local Lie group GL(2).(GL(2)is also a global Lie group, see Pontrjagin [ l ] . ) Let g(t), g(0) = e' be an analytic curve in GL(2) with tangent vector at e given by
We can identify
01
with the complex 2 x 2 matrix (1.11)
and L[GL(2)]= g42) with the space of all 2 x 2 complex matrices a. I n terms of this identification, an explicit calculation using (1.4) gives [ a , 81 = 4
-
8.
(1.12)
for a , /3 ~ g Z ( 2where ) the multiplication on the right-hand side of (1.12) is matrix multiplication. T h e special elements f + , 8-,f 3 , G,
(1.13)
1-2. EXAMPLES
7
form a basis for gZ(2) in the sense that every uniquely in the form a = a,$+
+ a&- +
.3y3
+
a,&,
a,
01
~ g Z ( 2can ) be written
, a2 , a3 , a4 E e'.
These basis elements obey the commutation relations
where 0 is the 2 x 2 matrix all of whose components are 0. - T h e 2 x 2 complex special linear group SL(2) is the abstract matrix group of all 2 x 2 nonsingular matrices
such that det g = 1. Clearly, SL(2) is a subgroup of GL(2). We can introduce coordinates for a group element g in a neighborhood of the identity e of SL(2) by setting g
+
= (g1 ,g, , g3) = ( a - 1, b, 4.
(1.16)
Here, d = (1 bc)/a. I n terms of these coordinates SL(2) is a 3-dimensional local Lie group. (It is also a global Lie group.) Suppose g ( t ) , g(0) = e, is an analytic curve whose tangent vector at e is a = (9? a2
a
,a 3 )
=
d
1, b ( t ) ,c(t))j
(a(t) -
t=o
;
can be identified with the complex 2 x 2 matrix (1.17)
since d
dt
i1 +
b(t)cft)
~
a(t)
Thus, L[SL(2)]= 4 2 ) is the space of all 2 x 2 complex matrices with trace zero. As in our first example, the Lie product is given by [a, /3] = a/3 - /3a for a, /3 E 4 2 ) . T h e elements
8
1. RESUMC OF LIE THEORY
satisfying the commutation relations [f3,
$+I
=
$+,
[f3 $-I ,
=
-Y-, [Y+, 2-1 = 2x3
(1.19)
form a basis for sZ(2). Since SL(2) is of great importance in special function theory we will study it in somewhat more detail than GL(2). For g E SL(2), a E sZ(2), Eqs. (1.8) can be written in the matrix form (1.20)
T h e solution of this equation is (1.21)
which converges for all t E @. Thus, in this case exp at = eat and the map a ---t e m is an analytic diffeomorphism of a neighborhood of 0 E $42) onto a neighborhood of e E SL(2). Some particular examples of this mapping which will be useful later are
Corresponding to any g E SL(2) define the inner automorphism ps of
SLCa bv ,
-
h E s~5(2),
d h ) = ghg-l,
(1.23)
where g-1 is the matrix inverse to g. By Theorem 1.1, pg induces an autornorphism p.g* of 4 2 ) where $(a) =
If a = xl$+
+ x2$- +
(1.24)
CtESE(2).
gag-', x3f3
=
(s,!'2 -x2
-xl
-x3/2
)
direct computation shows that the action of pp* on a is
ad -
+
bc x3 - bdx,
+ acx,
--abx3
+ b2x, - a2xl
-x3 + bcx,
' -bc
This equation will prove useful in Chapter 5 .
- acx1
.
(1.25)
1-2. EXAMPLES
9
The abstract group G(0,l) consists of all 4 x 4 matrices of the form a , b, c, 7 E @,
(1.26)
where the group operation is matrix multiplication. It is easy to verify that G(0, 1) is in fact a group. Indeed, g-l
=
and
1
-C
0
e-5 0
0
0
-U+bC
-T
!) E
G(0, 1)
(1.27)
where g, and g, are matrices of the form (1.26). We can introduce coordinates for the element g in G(0, 1) by setting ( a , b, C,
g
(1.29)
T).
Thus, G(0, 1) is a complex 4-dimensional Lie group. I n this case the coordinates (1.29) are valid over the entire group and not just in a neighborhood of the identity. The group G(0, 1 ) is said to be simply connected (Pontrjagin [13, Chapter 8). L[G(O, l)] can be identified with the space of 4 x 4 matrices of the form
a =
i" : : ") 0 0
0
0
0 0
0
0
with Lie product [a, /3]
= a/3
-
0 0 0 0 0 O 0 01,
.=(. .), 0 0 0 1 0 0 1 0 0 0 0 0 0
x1 , x2 1 x,
7
x4 E
@,
(1.30)
.-(. oj,
POI,a , /3 EL[G(O,l)].
f+=(o0
0 0 0 0
'
The matrices
0 1 0 0 0 0 0 0 0 0 0 0
0 E = (0 0 0
0 O 0 0
1 O 0 0
0
0) 0 0
(1.31)
10
1. RESUME OF LIE THEORY
with commutation relations
where 0 is the 4 x 4 zero matrix, form a basis for L[G(O, I)]. The exponential map exp a, a EL[G(O,l)], defined by Eqs. (1.8) takes the form
and in this case is an analytic diffeomorphism mapping all of L[G(O, l)] onto G(0, 1). In particular,
expTf3 =
0 0 1 0 0 0 0 1
(: oj,
0 0 0 1
.).
0 0 1 01 0
l O a O e x p a € = (o0 0 1 01 0
0 0 0 1
0 0 0 1
c
exPC%-=
exp by+=
o
o
a, b, c,
T
E
(1.33)
G'.
If gE G(0, I), the inner automorphism po of G(0, 1) defined by pp(h)= ghg-1, induces an automorphism p$ of L[G(O,l)] where &a) = gag-' for a EL[G(O,I)]. If a = x , f + x2$x3j3 x48,
+
direct computation yields p:(a) = ( x p
- x3b)/+
+
+
+ (x2e-' + x3c)f-
+ x 3 f 3 + (xIceT - x,be-l
- x3bc + x4)€,
(1.34)
where g is given by (1.26). We will make use of this equation in Chapter 4. The matrix group T3 is the set of all 4 x 4 matrices
1-2. EXAMPLES
11
T h e inverse of g E T, is given by
(1.36)
and the product g,g, by 0
71
+
72
(1.37) 1
where g, and g, are matrices of the form (1.35). We can establish a coordinate system for T , by assigning to g E T3 the coordinates g = (b, c, 7 ) .
(1.38)
T, is clearly a 3-dimensional complex local Lie group. Moreover, the coordinates (1.38) can be extended over all of Thus, T , has the topology of and is simply connected (Pontrjagin [l], Chapter 8). 9, = L( T3) can be identified with the space of matrices of the form
c3.
c3
o o o x , (1.39)
0
0
0
where the Lie product is [a, /3] is provided by the matrices
0 =
a/3 - Pa, a, ,0 E
;;?)I
0 0 0 0
8+=(:
0 0 0 0
8 - =(o0
0 0 o)l 1
0 0 0 0
0 0 0 0
(:-; ; :) 0
0
0
1
0
0
0
f3=
0
with commutation relations
. A basis for F3
(1-40)
12
1. RESUME OF LIE THEORY
T h e mapping a + exp a of F, onto T , , Eq. (1.8), is exp a Some particular cases of interest are 1
0
0
0
0
0
=
em.
1 0 0 0
1
.j.
0 0 0 1
(1.42)
1 0 0 0 0 1 0 exPC%-= (o 0 1
0 0 0 1
T h e map pz defined for g E T , by automorphism of F, . If a = x l f + (1.35) we have Cl-,(a) = (x,eT
1-3
-
bx,)$+
&(a)
=
gag-’ for all a E F, is an and g is given by
+ x2f- + x3f3
+ (x2ecT+ cx,)/- + x3Y3.
(1.43)
Local Transformation Groups
This section contains the principal results of the Lie theory of local transformation groups which are applicable to special function theory. I n particular, Theorem 1.10 will be used frequently in the remainder of this book. All the definitions and theorems will be stated for complex groups acting on complex manifolds; the results for real groups acting on real manifolds are completely analogous. Let G be an n-dimensional local Lie group and U an open set in Cm. Suppose there is given a mapping F: U x G + Cm and write F(x,g) = x g E C m f o r x . U , g E G . Definition G acts on the manifold U as a local L i e transformation group if the mapping F satisfies the conditions:
(1) xg is analytic in the coordinates of x and g; (2) xe = x; g, g‘ E G. (3) If xg E U then (xg)g’ = x(gg’), Here e is the identity element of G and X E U is designated by its coordinates x = (xl ,..., x,J. With the exception of the identity e, the elements of G have been printed in lightface type to distinguish them from elements of U .
1-3. LOCAL TRANSFORMATION GROUPS
13
If x E U and g is in a sufficiently small neighborhood of e, conditions (2) and (3) yield the relation (xg)g-' = x. Thus, the map x ~ x isg
locally one-to-one for fixed g. Let exp at, a E L(G), be a 1-parameter group in G. If xo E U we call the curve x(t) = xo exp at the trajectory of xo under exp at. The function x ( t ) can be expanded in a Taylor series in t about the point t = 0: or
where (1.45)
Thus, the vector with components
Qi=
c ajPjf(x), n
i
=
1,..., m,
(1.46)
j=l
is tangent at x to the trajectory of exp at through X. If f ( x ) is a function analytic in a neighborhood of some point xoE U, we define analytic functions [(exp a t)fl(x ) for a E L(G) and suitably small values of [ t I by [(exp atlfl(x) = f(x exp a t ) .
( 1.47)
Definition The Lie derivative Laf of an analytic function f ( x ) is
By direct computation we obtain
or n
La
=
c m
ajPj,(X)-
+l i=l
a axi
(1.48)
The commutator [ L aLa] , of the Lie derivatives L a ,L, is defined by [ L a La] = L.Jp - L&a . 9
14
1. RCSUME OF LIE THEORY
Theorem 1.5 (Lie's Second Fundamental Theorem) The set of all Lie derivatives of a local Lie transformation group G forms a Lie algebra which is a homomorphic image of L(G).I n fact, L + a g = aLa
+
4
3
9
Lca.gl = [La *L,l
for all a, b E @, a,9 , EL(G ) . If L(G) is isomorphic to the algebra of Lie derivatives of a local transformation group G we say G acts effectively. Given x E U and a E L ( G ) ,let x ( t ) be the solution of the system of equations i = 1 , 2,...,m,
(1.49)
with the initial condition x(0) = xo. The function x ( t ) is uniquely determined by a and is defined for suitably small values of 1 t I. Theorem 1.6 (Lie's First Fundamental Theorem) solution of the equation d -x dt
=Lax,
The unique
x ( 0 ) = xo
is the trajectory x ( t ) = xoexp at. Consequently, a knowledge of the Lie derivatives L, determines the action of G on U . T h e following result, which is of basic importance for the approach to special function theory presented in this book, states that any Lie algebra of differential operators is the algebra of Lie derivatives of a local Lie transformation group. Theorem 1.7 (Converse of Lie's Second Fundamental Theorem) Let j = 1,2,...,n,
be a set of n linearly independent differential operators defined and analytic in an open set U C e(m. If there exist constants C:k such that
1-3. LOCAL TRANSFORMATION GROUPS
15
the complex n-dimensional Lie algebra generated by the L, is the algebra of Lie derivatives of a local Lie transformation group G acting effectively on U. The action of G on xoE U is obtained by solving the equations d -x dt
=
(
x(0) = xo,
a j ~ x, j )
aj constant.
1=1
(The operators L, are said to be linearly independent on U if there exist no constants uj E @, not all zero, such that Ctf=, ujLjf = 0 for all analytic functions f on U.) I n this book we shall often be concerned with a variant of the last theorem. Let L j ,j = 1, 2, ..., n, be differential operators defined in an open set U C Cm+l and satisfying the hypotheses of Theorem 1.7. Suppose also,
a aXrn+l
+ 1,
i = 1 ,...,m
Pj,(x)= 0,
j = 1,...,n,
for all x E U . In other words,
a
[cI ,Lj
= 0.
T o emphasize the distinction between xm+l and the remaining components of x, we set xmtl = w and write x = (x‘, w )
where x’ = (xl,..., xm). According to Theorem 1.7, the Lj generate a Lie algebra which is the algebra of Lie derivatives of a local Lie group G acting on the open set U C gm+l.T h e action of G on U is given by integration of the equations d
-xi(t) dt dw
dt (2)
n
=
2 ajPji(x’), i=l
c n
=
j=1
ajpj
rn+l(x’),
i = 1,..., m, (1.50)
x(0) = xo.
Since the right-hand side of these equations is independent of w , the action of the 1-parameter group exp art on x takes the form x exp at = (x’ exp at, q(x’, exp at)
+w)
16
1. RESUME OF LIE THEORY
where q is a complex-valued analytic function of its arguments. T h e action of G on U is completely determined by the group elements exp at, so we have
xg
=
(x’g,dx’,g )
+ w),
g E G.
(1.51)
If xg E U , g’ E G, then (xg)g’ = x(gg’), which leads to the relations (x’g)g’= x’(gg’), or
dx’,gg’) = dx’,g) + q(x’g,g’),
4x’,gg’) = 4 X ‘ t 81 4x2,g’)
(1.52)
where v(x’,g) = Moreover, if g = e , then q(x’,e ) = 0, v(x’,e) = 1, for all x E U. Let LZw be the set of all functions f(x) on @+l, analytic in a neighborhood of some fixed point X O E U and such that f(x) = ewf’(x’),i.e., e-”f(x) is independent of w. Without loss of generality we can assume xo = 0 = (0,..., 0). Given g E G, define the mapping TB: LZw -+LZw as follows: TgfW =f(xg), f E a w * (1.53) Expression (1.53) is valid only for x in a sufficiently small neighborhood Moreover, for suitably restricted choices of g, g’ E G, x E U of 0 E @+l. and f E LZw , we have T”g‘f(x)= Tq(Tr‘f)(x). (1.54) Let LZ‘ be the set of all functions f’(x’) on @, analytic in some neighborhood of 0’ = (0,..., 0). There is a one-to-one mapping p of LZ‘ onto LZw given by f ’ E a‘. p : f’(x’)+ ewf‘(x‘), Since this mapping is invertible it is easy to see that the operators TQ on LZw induce operators T’g = p-1Tgp on GY with the following properties: (1) [T’”’l(X’) = +‘, g)f‘(x’g), (2) [T’‘f’](x’)= f ’ ( ~ ’ ) ,
(1.55)
’1 (x’) = [T’gl(T’”zf’)](x’), (3) [T’Q~gzf valid for all f‘ E a’, x‘ E @m, g, g, ,g, (1.55) make sense.
E
G such that both sides of Eqs.
Definition Let G be a local Lie transformation group acting on an 0 E U , and let LZ be the set of all complexopen neighborhood U of
cm,
1-3. LOCAL TRANSFORMATION GROUPS
17
valued functions on U analytic in a neighborhood of 0. A (local) multiplier representation T.of G on 6Z with multiplier v, consists of a mapping P(g) of Gl onto 6Z defined for g E G, f E Gl by xE
[ T w f l ( x ) = 4x9 g)f(xg),
where
V(X, g)
u 9
is a complex-valued function analytic in x and g, such that
(1) v(x,e ) = 1, all X E U , (2) v(x,g1g2) = v(x, g,)v(xg, g2), 9
g1 gz g1g2 E G. 9
Property (2) is equivalent to the relation rW,g2)fl(x)
-
I ~ ( g , ) ( T ( g J11 f (x).
Thus, the mappings T’ given by Eqs. (1.55) define a multiplier representation of G. Furthermore, every multiplier representation of G arises in this way. T hat is, if T.is a multiplier representation of G on U C grn we can define an action of G as a transformation group on a neighborhood of @+I by ( x , w) -+ (xg,w ln[v(x, g)]) where x E U and w is a complex number. T h e mapping T’(g)induced by this action is just P(g). Let T.be a multiplier representation of G and let f E Gl. If 01 E L ( G ) then Tu(exp a t ) f E GZ for sufficiently small values of 1 t j. Definition T h e generalized Lie derivative D,f of an analytic function f ( x ) under the 1-parameter group exp at is the analytic function
(1.56)
For v = 1 the generalized Lie derivative becomes the ordinary Lie derivative. By direct computation from (1.56) we obtain
are defined by (1.45) and the analytic where the analytic functions Pi,(x) are defined by functions Pj(x)
I n analogy with Theorem 1.5 we have:
1. RESUME OF LIE THEORY
18
Theorem 1.8 T h e generalized Lie derivatives of a local Lie transformation group form a Lie algebra under the operations of addition of derivatives and Lie product [D, D,I 7
=
D,D,
-
D,D,
'
This algebra is a homomorphic image of L(G): D,,,
=
D,
+ D,
7
D,,,,
=
Da,
[D, Do], 3
=
aD,
-
Definition A Lie group G acts effectively in a local multiplier representation TY if L(G) is isomorphic to the algebra of generalized Lie derivatives. Making use of the relationship between local transformation groups acting on Cm+l and multiplier representations on Cmpresented above we can easily prove the following analogies of Theorems 1.6 and 1.7: Theorem 1.9 A local multiplier representation TY of G is completely determined by its generalized Lie derivatives. Theorem 1.10
Let
1 Pj,(x)ga + Pj(x), m
Dj(x)=
i = 1,..., n,
i=l
be a set of n linearly independent differential operators defined and analytic in an open set U C Cm. If there exist constants C,!k such that n
[Dj , Dk] = DjDk - DkDj =
CikD,,
1 <j,
k
1=1
< n,
then the complex linear combinations of the D j form a Lie algebra which is the algebra of generalized Lie derivatives of an effective local multiplier representation TY. T h e action of the group is obtained by integration of the equations
d
n
dt xi(t)=
C ajPji(x(t)),
j=1
~ ~ (= 0 )x i ;
i
=
1, 2,..., m, (1.58)
d dt
-v(xo,
n
exp at) = v(x0, exp a t )
where x ( t ) = xoexp cut, a EL(G).
1 ajPj(x(t)),
3=1
v(x0, e)
= I,
1-4. EXAMPLES OF LOCAL TRANSFORMATION GROUPS
19
According to this theorem the action of the 1-parameter group exp at on f E a is given by [TY(exp at)f](xO)= v(xo,exp at)f(xO exp at)
(1.59)
where v and x ( t ) are the solutions of Eqs. (1.58). Since G is completely determined by the elements exp a , a EL(G), Eqs. (1.59) completely determine TY. Moreover, [T.(exp at)f](x)is an analytic function of t for I t 1 sufficiently small. By direct computation, dn dt
-[TTexP .t).fl(x)l
t=o
=
[D,nfl(X)
(1.60)
where D: = DUDU ..Be( n times). Expansion of [T.(exp at)f] (x) in a Taylor series proves Theorem 1.1 1 [TY(exp at)f](x) =
t" 1[Dzf](x). O0
n-1
Theorem 1.11 is another justification for the use of the exp notation. Let T " k be a multiplier representation of the local Lie transformation group Gk acting on an open neighborhood U , of Qrn for k = 1,2, respectively . Definition T h e multiplier representations T"1, T"2 are isomorphic if there is an analytic isomorphism p of G, onto G, and an analytic one-to-one coordinate transformation ~p of a neighborhood of 0 E U, onto a neighborhood of 0 E U , such that
'p(x1g1) = cp(xl>P.(gl), (2) v2(Cp(x1), p.(g1)>= v 1 b 1 g1)
(1)
7
for x1 E U, , g, E G, . Ordinarily we shall not distinguish between isomorphic multiplier representations.
1-4
Examples of Local Transformation Groups
I n this section the fundamental theorems on local transformation groups will be illustrated by means of examples. Furthermore, a brief indication of the relationship between each example and special function theory will be given. T h e next four chapters constitute a detailed and systematic study of the relationship between Lie groups and special functions which is indicated here.
20
1. RESUME OF LIE THEORY
Consider the differential operators J+, J-, 53
=
--
d u+z--, dz
J+
+
=~ U Z
2'-
J3
defined by
d
dz '
J-
==
d
--
dz '
(1.61)
where z is a complex variable taking values in some neighborhood U of 0 E @, and 2u is a nonnegative integer. These operators satisfy the commutation relations
[J3, J']
=
[J3, J-]
J+,
=
-J-,
[J+, J-I
=
2J3
which are identical with the commutation relations (1.19) for the generators of L[SL(2)]= sl(2). Thus, by Theorem 1.10 the operators J+, J-, J 3 generate the Lie algebra of generalized Lie derivatives of a local multiplier representation TY of SL(2).We will explicitly verify this and compute the multiplier V. According to (1.58) the action of the 1-parameter group exp T j 3 , T E @, is obtained by integration of the differential equations d
d7z=z,
d
dT v(zo, exp ~ -
3= --uv(zo, ~ )
exp T f s )
(1.62)
with initial conditions z(0) = zo, v(zo,e ) 3 I. T h e solution is easily seen to be Z ( T ) = zoeT, v(zo,exp Ty3)= cUT. (1.63) Thus, iff
E Ul
is analytic in some neighborhood of zo E U then [TY(exp T f 3 ) f ] ( Z o )
for
= e-u7f(zoe7)
(1.64)
1 T 1 sufficiently small. Similarly, we obtain
defined for
REMARK T o give a precise meaning to Eqs. ( .64) and (1.65) it would be necessary to state explicitly the values of T, b, c, zo and the functions f for which the equations are defined. However, these values are easily determined by inspection once the domain o ff is given and ordinarily we will not bother to write them down. Similar remarks hold for all the computations involving local transformation groups considered in this book.
1-4. EXAMPLES
OF LOCAL TRANSFORMATION GROUPS
21
Equations (1.64) and (1.65) completely determine TY. T o show this we use (1.22) and compute the product
Thus, if
is in a sufficiently small neighborhood of the identity we can write g uniquely in the form g
=
exp b'$+ exp c'$-
where er'l2 = d-I, 6' = -b/d, c' f E GZ is then given by
=
exp 7'y3
(1.67)
-cd. T h e operator F(g) acting on
[ ~ ~ g =i [ Wm ~ P b ~ + ) m e CY-) x ~
WXP
7 ' 9 3 1 m )
(1.68)
+
+
and we have v ( z , g ) = (bz + d)211,zg = ( U Z c)/(bz d), defined for z and g in suitably small neighborhoods of 0 and e, respectively. I t is easy to verify directly that Eq. (1.69) defines a local multiplier representation of SL(2)and that the differential operators J+, J-, J3 are generalized Lie derivatives of T.. Now that we have constructed the multiplier representation (1.69) we will indicate its relation to special function theory. Let V'-(u)be the vector subspace of G! consisting of all complex linear combinations of
22
1. RESUME OF
LIE THEORY
the 2u + 1 functions h,(x) = 21, I = 0, 1,..., 2u. I t follows from (1.69) ) Tu(g)fEV ( u ) . Furthermore, for f~ Y'-(u) the rightthat i f f € V ( uthen hand side of (1.69) is defined for all g E SL(2),not just for g in a suitable neighborhood of e, and TY(g,g,)f = T(g,)[TY(g,)f] for all g, ,g2 E SL(2). T h e matrix elements Dzk(g)of P with respect to the basis functions h , are defined by
c Dz,(g) hz(z), 2U
[TY(g)hJ(z)
=
z=o
g 6 =(2),
k
=
0 , I,..., 2%
(1.70)
or, explicitly, (ax
+ c ) ' ( ~ z+ d)'"-'
221
=
C D&)z'
(1.71)
l=O
(z
where g = i). Since TY is a multiplier representation of SL(2) the matrix elements satisfy the addition theorems
Using the binomial theorem to expand the left-hand side of (1.71), we obtain the following expressions for the matrix elements D,,(g):
if
2u>k>l>O,
(1.73)
if 2 u > l > k > O .
Here, F is a hypergeometric polynomial, see (A.4). Thus, the matrix elements are well-known special functions and Eqs. (1.72) yield addition theorems for these functions. As a second example consider the differential operators Jf, J-, J 3 defined by (1.61) where now 2u E @ is not a nonnegative integer. I n exact analogy with the preceding example, these operators induce a multiplier representation TY of SL(2) defined by (1.69)'
where z and g are in suitably small neighborhoods of 0 and e, respectively. Since 2u is not a nonnegative integer, the multiplier v ( z , g ) =
1-4. EXAMPLES OF LOCAL TRANSFORMATION GROUPS
23
+
( b z d)2uis defined by its power series expansion in z about z = 0. This power series converges only if I bz/d 1 < 1. Furthermore, if f E GZ and z is in the domain off then
is in the domain o f f and 1 az/c 1 < 1, I bzld I < 1. There is now no finite-dimensional vector subspace of polynomials in z which is invariant under (1.69)', so we consider the action of TY on all of GZ. Since the elements of GZ are just those functions with convergent power series expansions about z = 0, the action of TY on GZ can be determined from I = 0, 1 ,2 ,... . Thus, a knowledge of TY on the basis functions h z ( z )= zz, we define the matrix elements B,,(g) with respect to the basis functions (1.74)
(1.75)
Since TY is a multiplier representation we have (1.76)
for g, ,g, in a sufficiently small neighborhood of e . T h e precise domain of validity of the addition theorems (1.76) will be determined in Chapter 5. T h e matrix elements can be computed by expanding the left-hand side of (1.75) as a power series in z and calculating the coefficient of zz. T h e result is
if k & Z > O ,
Bzk(g)
r
+
i
&d2u-'bZ-'T(2u - k 1) F -k, -2u r ( 2 u - I + 1)(1 - k ) ! if Z > k > O ,
+ I; I
-
k
+ -)
(1.77)
bc 1; ad
where is the gamma function and F is the hypergeometric function (see the appendix). This is an example of a multiplier representation of
24
1. RtSUME OF LIE THEORY
SL(2)that is purely local; it is defined only in a neighborhood of e, not on all of SL(2). T h e connection between addition theorems for special functions and matrix elements of local multiplier representations presented here has general validity. I n fact, we will show that the basic addition theorems for hypergeometric functions can be obtained from the study of multiplier representations of SL(2),the basic addition theorems for confluent hypergeometric functions follow from the study of multiplier representations of G(0, I), and addition theorems for Bessel functions can be obtained from multiplier representations of T,.
CHAPTER
2 Representations and Realizations of
Lie Algebras
I n Chapter 2 we establish a fundamental relationship between Lie groups and certain special functions: Special functions appear as basis vectors and matrix elements corresponding to local multiplier representations of Lie groups. T h e correspondence is sketched in Section 2-2, then illustrated by the familiar example of angular momentum operators and spherical harmonics. T h e 4-dimensional Lie algebras 9 ( a , b) are introduced in Section 2-5. Later, it will be shown that the representation theory of 9 ( a , b) corresponds to a study of special functions of hypergeometric type. As preliminary material toward the demonstration of this fact we classify the abstract irreducible representations of 9 ( a , b) in Section 2-6, and construct realizations of 9 ( a , 6) by means of generalized Lie derivatives in one and two complex variables in Sections 2-7 and 2-8.
2-1
Representations of Lie Algebras
I n this section the concept of a representation of a Lie algebra on an abstract vector space will be introduced. I t is assumed that the reader is familiar with the basic definitions and results of the theory of vector
26
2. REPRESENTATIONS OF LIE ALGEBRAS
spaces (Halmos [l], Dunford and Schwartz [l]). However, since most elementary textbooks on linear algebra are concerned primarily with finite-dimensional vector spaces it may be useful to review some of the fundamental definitions in a wide enough context to apply to infinitedimensional spaces. Infinite-dimensional abstract vector spaces will appear frequently in this book. Let V be a vector space over the field F. (Here, F will always be either the real numbers R,or the complex numbers, @.)We denote the additive identity of V by 0 and the additive identity of F by 0. A subset S of V is said to be linearly independent over F if for every finite subset v1 , w 2 ,..., w, of distinct elements of S and every sequence (a, , a, ,..., a,) of elements of F the equality C?=, u p i = 8 implies a, = a, = = a, = 0. A nonvoid linearly independent subset B is called a basis of V if it is maximal, i.e., if B is not a proper subset of some linearly independent set in V. It can be shown using Zorn's lemma that every vector space with at least two elements contains a basis (Dunford and Schwartz [l], Chapter 1). Furthermore, if B is a basis for V then any v E V can be written uniquely as a finite linear combination of elements of B: v = alwl
+ azwz + + unw, , *.*
a,
,...,a, E F ,
v1 ,..., vn E B.
T h e cardinality of a basis depends only on V and is called the dimension of V. If B contains m elements, V is rn-dimensional. If B contains an infinite number of elements, V is infinite-dimensional. (Similarly, if g is a Lie algebra over F a basis of 2'2 is a subset Y which is a basis of 59 considered as a vector space. The dimension of Y is the cardinality of a basis.) A linear operator on V is a mapping T : V -+ V such that
T h e product T,T, of two linear operators T , and T , on V is the linear operator defined by (T,T,)(v) = Tl(T,(v))for o E V. The sum T, T, is the linear operator defined by ( T , T,)(a) = T,(v) T2(w). If a E F and T is a linear operator we define the scalar multiple aT of T by (aT)(v)= aT(w) for all w E V. The set of all linear operators on V with the operations of product, sum, and scalar multiple forms an algebra. Moreover, if we define the commutator [ Tl , T,] of the linear operators T,, T , by [ T , , T,] = T,T, - T,T, , the operators on V together with the commutator [.,.I form a Lie algebra 9(V). Let Y be a Lie algebra over F.
+
+
+
2-2. REALIZAT1ONS OF REPRESENTATIONS p:
27
Definition A representation of 9 on V is a homomorphism 9 ---t 9( V). That is, p satisfies the conditions
(1)
p(a)~9(V)
(2) (3)
p(L.9 p(aa
PI)
=
+ bP)
for all a € 9,
b(4,P(P)19 = up(.)
+ b(P),
a, b EF, a,p E
9.
A subspace W of V is said to be invariant under p if p(a)w E W for all a E 9,w E W. A representation p of 9 on V is irreducible if there is no proper subspace W of V which is invariant under p. Two vector spaces V, V' over F are isomorphic if there exists a mapping p of V onto V' such that (1)
+
+
a2021 = alP(%) a2A4 for all a, , u2 E F, vl , v2 E V. (2) If p ( v) = 8' then v = 8. (3) For every v' E V' there exists a er E V such that p(v) = v'.
p(a1v1
Let p, p' be representations of 9 on V , V', respectively. p and p' are said to be isomorphic if there exists an isomorphism p of V and V' such that p(p(a)v) = p'(a)p(v) for all a E 9,v E V.
2-2
Realizations of Representations
Let G be a local Lie group with Lie algebra L(G) and suppose p is a representation of L(G) on the abstract vector space V. In analogy with the theory of local transformation groups presented in Chapter 1, it would seem natural to construct a mapping 6 of G into 9(V) as follows:
where a EL(G),v E V , and t E @. Unfortunately, the right-hand side of (2.1) may not be defined as an element of V since it involves an infinite
linear combination of vectors. Thus, expression (2.1) may be meaningless and the analogy between Lie algebras and local Lie groups breaks down. However, suppose p satisfies
CONDlTlON (A) V can be realized as a vector space whose elements are functions analytic in a neighborhood of some point zoE Crnand such that the operators ~ ( c YCY) ,E L ( G )are differential operators analytic at 2".
28
2. REPRESENTATIONS OF LIE ALGEBRAS
T o be more precise, let Ulzo be the set of all functions analytic in some neighborhood of zo(the germs of functions at zo, see Gunning and Rossi [l], Chapter 2). a,, can be given the structure of a complex vector space in the usual manner. Assume the existence of an isomorphism p of the abstract vector space V onto a subspace V of Ul,, . Then, the operators p’(0r) on V induced by the operators p(a), d ( G ) on V (p’(a)p(z,) = plp(a)z,],z, E V) obviously define a representation of L(G). Second, assume there exists a representation of L( G)by generalized Lie derivatives D, on Ule such that D, = p ( a ) on V for all (Y EL(G).If these two hypotheses are satisfied the vector space V can be considered as a subspace of GZzoand the operators p(a) can be identified with the generalized Lie derivatives D, . Under these circumstances Theorem 1.10 (or a slight modification of it; see Section 8-1) states that the representation p of L(G) induces a multiplier representation TY of G on a,, . I n particular, Eq. (2.1) is now well defined since the right-hand side of the equation is an infinite sum of analytic functions which converges to an element of Ole for I t I sufficiently small. (Here, the topology on GZzois the usual one of uniform convergence on compact sets of Qm.) According to Theorem 1-10, $(exp tor)z, = TY(exp ta)w for all E L ( G ) ,e, E V , and I t I sufficiently small. As noted above, V is invariant under the operators D, but not necessarily invariant under the operators ‘P(g), g E G.T o remedy this we extend V C Ule to a larger subspace 7 which is invariant under TY. Namely, we define P to be the intersection of all subspaces W of az0 which obey the conditions: W 3 Y and TY(g)w E W for all g E G, w E W such is the smallest subspace of aewhich that TY(g)w is defined. Clearly, contains V and which is invariant under TY. T h e elements of 4 are just the finite sums of elements of the form TY(g)o, where g E G,z, E V . Consequently, if condition (A) is satisfied a representation p of L(G) will induce a multiplier representation TY of G which leaves invariant. I n general, TY depends critically on the dimension m of Crnand on the choice of V . T o preserve the one-to-one correspondence between local Lie groups and Lie algebras, we need to find conditions which guarantee that the action of TY on P is in some way uniquely determined by p. If o1,..., ok ,... is a basis for V , the functions p(vl), p(oZ),..., p(‘uk),-.. form a basis for V . We say that the { p ( w k ) } form an analytic basis for 9 if every element u of Y can be expressed uniquely as a countable linear combination of the basis functions p(vk),uniformly convergent in some neighborhood of zo. Furthermore, it is required that the coefficients in this expansion be bounded linear functionals of the argument u (in the topology of uniform convergence on compact sets), i.e., if u =& ck(u)p(vk) then ck(u)+ 0 as u --t 0. (Y
(Y
2-2. REALIZATIONS OF REPRESENTATIONS
29
From Eq. (2.1) we obtain
~ E L ( G ) k,
=
1,2,...,
(2.2)
for I t 1 sufficiently small. If {p(ok)}is an analytic basis the right-hand side of (2.2) converges to an element of Q which is a (possibly infinite) linear combination of the functions p(vk). I n terms of this basis the matrix elements T,,(exp ta) are uniquely defined by m
TY(exp t a ) p ( v k ) =
k = I , 2,... .
1Tlk(expt a ) p ( v l ) ,
1=1
(2.3)
In fact, Eqs. (2.2) show that the matrix elements are uniquely determined by p for g in a sufficiently small neighborhood of e E G and are independent of the choice of Crnand V . Thus, restricting ourselves to vector spaces V of analytic functions which satisfy CONDlTlON (8) p(ol), p(w,) ,..., p(wk),..., is an analytic basis for 7,
we find that the matrix elements of the operators F(g) depend only on
p and the chosen basis of V; not on V . Let G, p, I/, {wk}, p, V be given such that conditions (A)and (B) are satisfied. Clearly, p(wl), p(w2),.,,,is an analytic basis for 7". Using this
analytic basis and (2.3) we can easily derive the following addition theorem for the matrix elements:
defined for g, ,g, in a suitably small neighborhood of e. T h e relation TY(g)p(wk) =
ct
Tlkk)
P(wl),
G,
?!
(2.3)'
1=1
can be regarded as a generating function for the matrix elements T I P ( gand ) the basis functions p ( v I ) . Finally, the equations D - P ( Z ) ~=) &(a)
nk),
a €L(G),
k
=
1,2,***,
(2.5)
can be interpreted as differential recursion relations for the functions p(wk).
30
2. REPRESENTATIONS OF LIE ALGEBRAS
The connection between these results and special function theory is now apparent. The matrix elements T&) are functions on the local group G and, if we make the proper choices for G and a basis of V, they will turn out to be familiar special functions. Moreover, the analytic functions p(qJ will often be expressible as special functions. I n this case Eqs. (2.4), (2.3)’, and (2.5) constitute addition theorems, generating functions, and recursion relations for the special functions T,,(g)and p(vk). I claim that a significant portion of special function theory is contained in these three sets of equations. Much of the remainder of this book will be devoted to the documentation of this claim by means of explicit computations. (Recall Section 1-4 where it was shown that the Jacobi polynomials appear as matrix elements of local multiplier representations of SL(2).) T o systematically study the special functions related to a given Lie algebra L(G) we could proceed as follows: (1) Classify “all” representations p of L(G). This is primarily an algcoraic problem. In order to obtain practical results it will ordinarily be necessary to study only those representations which possess certain convenient properties. In particular, we shall cla ;sify only the irreducible representations of L(G). (2) Classify “all” realizations of L(G)by generalized Lie derivatives. This is a problem in classical Lie theory which will be studied in Chapter 8. The basic difficulty here is how to decide when two realizations of L( G)by generalized Lie derivatives are “essentially” the same. (3) For each representation p of L(G) and each realization of L(G) by generalized Lie derivatives D, , find a realization 9 ‘ - of V such that conditions (A) and (B) are satisfied. (4) Choose a “suitable” basis in I‘ and compute the matrix elements T,,(g) and basis functions p(vk). For the low-dimensional Lie algebras considered in this book it will ordinarily be clear which basis in V should be chosen. The above description of a relation between Lie algebras and special functions is general and heuristic. T o show that this procedure leads to practical methods for uncovering the group structure of special functions we will henceforth be very specific. In particular, we will examine in detail the special functions associated with the Lie algebras L(SL(2)), L(G(0, l)), and L( T3).The special functions will turn out to be hypergeometric functions, confluent hypergeometric functions, and Bessel functions. These familiar functions will occur both as matrix elements and as basis vectors for irreducible representations of the above Lie algebras.
2-3. REPRESENTATIONS OF L(0J 2-3
31
Representations of L(0,)
Perhaps the best known example of the relation between Lie groups and special functions presented in Section 2-2 is the connection between the rotation group and spherical harmonics. Although this example is treated in most books on quantum mechanics, it may be instructive to examine it again to see how it fits into the general framework of the last section. It is a remarkable fact that essentially the same methods which are used to derive the fundamental properties of spherical harmonics from the representation theory of the rotation group, suffice to derive the fundamental properties of hypergeometric, confluent hypergeometric and Bessel functions. The rotation group 0, in 3-dimensional space is the group of real 3 x 3 matrices A such that A A L= I and det A = + I (Hamermesh [I]). Here A t is the transpose of A and I is the 3 x 3 identity matrix. 0, is a real 3-parameter Lie group. The real Lie algebra L(0,) has a , , y 3with commutation relations basis
x2
Let p be a finite-dimensional irreducible representation of L ( 0 3 ) on the complex vector space V and define the operators J1 , Jz , J3 on V by p($?jk) = J k , k = 1, 2, 3. These operators generate a Lie algebra which is a homomorphic image of L(0,). However, for many purposes it is more convenient to use the operators J+, J-, J3: J3
= iJ,,
J+ = -J,
+ iJ1,
J- = J,
+ iJ,
(i =
a) (2.7) .
The commutation relations become
[J3, J+l
=
J+,
[J3, J-I
=
-J-,
[J+, J-I
= 2J3
(2.8)
where now [A, B] = A B - BA. Note that (2.8) is formally identical to Eq. (1.19) for the commutation relations of the generators of the complex Lie algebra 4 2 ) . T o determine all finite-dimensional irreducible representations of L ( 0 , ) it is sufficient to classify (up to isomorphism) all nonzero complex vector spaces V and operators Jf, J-, J3 on V satisfying ( 2 4 , such that no proper subspace of V is invariant under J+, J-, J3. Let V be one such vector space and let h, E V be a nonzero eigenvector of J3 with eigenvalue q: J3h, = qh,.
2. REPRESENTATIONS OF LIE ALGEBRAS
32
+
T h e relation [J3, J+] h, = J+h, implies J3(J+h,) = (q 1) J+h, . Hence, either Jfh, = 0 or J+h, is an eigenvector of J3 with eigenvalue q 1. Similarly it is easy to show that either J-h, = 0 or J-h, is an eigenvector of J3 with eigenvalue q - 1. By repeating the above argument we find J3(J+)'ha
= (4
+ k)(J+)"ha
J3(J-)"g
x=
+
( 4 - k)(J-)kh,
for all positive integers k . Since V is finite-dimensional there must
exist an integer Y 3 0 satisfying (J+)'h, # 0 and (J+)'+'h, = 0. Set = f i where 1 = q Y , so that J"fr = r f i . By a similar argument there is an integer s 3 0 such that (J-)"fi # 0 and (J-)s+'fi= 0. We will show that the vectors f k , k = 1, 1 - 1,..., 1 - s, defined by fZpj= ( J-)ifi , j = 0, ..., s, form a basis for V . One way to demonstrate this is by means of the operator Cls0= J+JJ3J3 - J3 (the Casimir operator). As can easily be verified from the commutation relations (2.8), Cl,o commutes with J+, J-, and J3:
+
(J+)'h,
+
[Cis,
9
11'
=
+
J3J?fi
[Ci,o
J-I
=
[C,,o
9
J31
(2.9)
= 0.
Moreover, Ci.ofi
=
J'J-fi
= (I-J'
so Cl,ofz= Z(Z J- we have
- JYi
+ 2J3)fi + (12 - l ) f i
+ l ) f i , because
= J-J+fi
+ l(l +
1)fi
;
J+fz = 0. Since Cl,o commutes with
Cl,ofi--, = Ci,o(J-)'j~ = (J-1' Ci,ofi
+
10 + l)(I-)'fi
= l(1
+ 1)fl-j
for j = 1, 2 ,..., s. Thus, Cl,ofk = l(1 l)fk for k = 1, 1 - 1,..., 1 - S. On the other hand we can evaluate Cl,ofi-, directly: Ci.ofi--s
= (J'J-
+ J3J3 - J'))i-s
= ( I - s)(l - s -
1)fi-8
9
since J-fi-, = 0. Comparison of these two results yields 1(1+ 1) = (1 - s)(I - s - 1) or s = 21. Since s is a nonnegative integer, 1 is either an integer or half an integer. Another direct computation proves C1,ofi-j
=
J+(J-)"'fi
=
J+fi-j-1
+
J3J3fi-j
+ (1 - j ) ( l
- J"fi-9
-j -
+
1)fi-J
for j = 1 , 2,..., s - 1 . From Cl,of i V j = Z(l l)fl-j we obtain 1) fk+l for k = 1 - 1, 1 - 2 ,..., -1. These J+fk = ( I - k)(I k results show that the (21 1)-dimensional subspace of V generated by the vectors f k , k = 1, 1 - 1,..., -1, is invariant under J+, J-, and J3.
+ +
+
2-4. THE ANGULAR MOMENTUM OPERATORS
33
Moreover, this subspace must coincide with V itself because V is irreducible under p. We know how the operators J+, J-, J3 act on the basis vectors fk of V , so p is completely determined: J”fk
= kfK 9
J-fk
= fk-i
J + ~ E= ( I
,
+
K
=
-
+k+
l)fk+i
9
(2.10)
I, I - 1,..., -I.
I(I l ) f k , (On the right-hand sides of Eqs. (2.10) we make the convention:fk = 0 if k is not in the spectrum of J3.) Conversely, for any nonnegative integer 21 it is easy to verify that the operators J+, J-, J3 on V defined by (2.10) satisfy the commutation relations (2.8) and thus determine an irreducible representation D(21) of 0,. The irreducible representation D(21) is uniquely determined by and uniquely determines the spectrum of the operator J3 in this representation, i.e., the eigenvalues {+l, 1 - 1,..., -I>. However, the basis vectors {fk} are not uniquely determined by D(21). In particular, if {Yk}, k = 1,..., -1, is a set of nonzero complex constants the vectorsf; = r k f k will also form a basis of V consisting of eigenvectors of J3. A most convenient basis for V is obtained by choosing the constants Yk such that yk+Jyk = [(I k 1)(1 k = 1 - 1, 1 - 2,..., -1. For the new basis vectors {f:}, relations (2.10) become C,,,f,
=
+ +
JY; J+fk
= kf;,
=[(l-
kMI+
J-j;
=
[(E
+ k)(I- k + l)]l’Yk-l,
+ l)ll’Y;+l,
C1,,fk
= I(/+
1)f;
-
(2.11)
Relations (2.11) are especially convenient for the study of unitary representations of 0, (see Section 5-16). From the above discussion we can conclude that the only finitedimensional irreducible representations of L( 0,) are the representations 0(21), 21 a nonnegative integer. The representation space corresponding to D(21) has dimension 21 1 and contains a basis fi , fi-l ,...,fllsuch that the action of J+, J-, J3 on the basis is given by (2.11).
+
2-4 The Angular Momentum Operators In the study of the quantum theory of angular momentum the differential operators
a ax,
J, = x , - - x
-,a
ax,
J, = XZ
a ax, a - XIa JZ=X~--X
ax1
8x2
a
- 7
ax,
(2.12)
2. REPRESENTATIONS OF LIE ALGEBRAS
34
occur, where x, , x, ,x, are real Cartesian coordinates (Landau and Lifshitz El], Chapter 4). These operators satisfy the commutation relations (2.6) and thus generate a Lie algebra isomorphic to L (0 ,). I n terms of the spherical coordinates I, 0, p' defined by x1 = r cos 8 cos p,
I>o,
x3 = r sin 0,
x p = T cos 3l sin p,
o
~
e
O~ < ~~ P ,, ~ ~ ,
the operators J+, J-, J3 take the form
2) a(P
J3=-i-.
a
av
'
(2.13)
T o relate the above results to special function theory we look for a realization of the representation D(21) such that the basis space Y is a space of analytic functions of the real variables 8, cp and the operators J+, J-, J3 are given by (2.13). Thus, we try to find the functions YF(0, p') such that
+ k)(l - k + l)]l/'Yf-', J+YL = [(I - k ) ( l + K + l)]1/2Yy1, C,,,Yf = 1(1 + l)Yf, J'Yf
= kYf
,
J-Yk
= [(I
(2.14)
1
k From
J3 =
--i
a/+
= 1,l - 1,..., -1.
there follows the relation
so Yf can be written in the form Yf(0, cp) = Pf(0)e"r. The problem of finding functions Y: satisfying (2.12) thus reduces to the problem of determining the functions P:. Furthermore, the requirement J+Yt = 0 leads to the differential equation
dp: - 1 cot e P; = 0 dd
with solution Pt(0) = cLsin' 0
= c,(l - cos2
e)z/z
where c, is an arbitrary nonzero constant. The functions Pfcan now be defined recursively from Pi by the condition
2-4. THE ANGULAR MOMENTUM OPERATORS
35
By induction, we obtain the explicit expression
k The requirement J-Y;'
= 1,l - 1,..., -1.
(2.15)
= 0 leads to the condition
This condition is satisfied only if 1 is an integer; if 1 is not an integer the constructions fails. I n case 1 is an integer the functions f; = Y f ,k = 1 , l - 1,..., -1, form a basis for the representation D(21) of L(0,). T o see this, note that we have mimicked the abstract construction of the representation D(21) given in Section 2-3. Thus, we have found a function f; = Y : such that J+fi = 0 and have determined the functions fi defined by
f; = Yf =
[(21)!(1 (+I k)!k ) ! l -
k
(J-)"'Y;,
= 1, 1 -
1,..., -1.
Since J-Yi' = J-fLr = 0, the argument in the previous section proves that the basis functions satisfy Eqs. (2.1 1). That is,
+
+ + +
d K)(I - K dB d - Pf - k cot 0 Pf = [(I - k)(l k dB
- -P; - K cot e P; = [(I
k
=
1)11/2 ~ f - 1
1)]1/2
Pf+1
(2.16)
1, 1 - 1 ,..., -1,
where the last equation is obtained by writing Cl,of; = 1(1+ I ) f; in terms of the differential operators (2.13). The arbitrary constant c I in Eq. (2.15) is usually fixed by the requirement
,:"1: 1 Y:(O,v)i2
sin 0 d0 d v = 1.
Evaluating this integral we find
where the phase factor (- 1)' is introduced to conform to convention.
2. REPRESENTATIONS OF LIE ALGEBRAS
36
T h e functions Y:(O,p’) = P;(O)& are known as spherical harmonics and Eqs. (2.16) give fundamental recursion relations and a differential equation for these functions (ErdClyi et al. [l], Vol. 11). Thus, if I is a nonnegative integer the spherical harmonics Y : form a basis for the representation D(21). Although we have failed to find a realization of those representations for which 1 is not an integer, in Section 5-14 we will find such realizations by choosing homomorphisms of L(0,) by generalized Lie derivatives different from (2.12). The representations of L(0,) realized here can be extended to local multiplier representations of 0, . T h e matrix elements can be computed and the results yield generating functions and addition theorems for the spherical harmonics. These results are well known (Gel’fand et al. [ 13) and will be included in Section 5-16. T h e methods used to relate spherical harmonics to representations of L(0,) are applicable to a wide variety of special functions. I n the next section a family of Lie algebras 9 ( a , b) will be introduced and the irreducible representations of these Lie algebras will be classified, subject to suitable restrictions. We will find realizations of the irreducible representations in terms of generalized Lie derivatives acting on spaces of analytic functions. For each such realization there will exist a natural basis of special functions. T h e special functions so obtained are the hypergeometric, confluent hypergeometric, and Bessel functions. This relation between Lie algebras and special functions provides insight into special function theory.
2-5
The Lie Algebras 9(0,6)
For any pair of complex numbers (a,b) define the 4-dimensional complex Lie algebra 9 ( a , b) with basis $+, $-, $,, 8 by
[$+, 9-1=
- b&,
[f3,
$+I
=
$+,
[f3,
9-1= -$-
(2.17)
[ f +81 , = [8-, 61 = [ f 3 , 81 = 8, is the commutator bracket and 0 is the additive identity
where [.,.I element. (It is easy to verify that relations (2.17) do in fact define a Lie algebra.) For special choices of the parameters a, b, %’(a,b) essentially coincides with one of the Lie algebras introduced in Section 1-2. I n particular we have the following isomorphisms: cg( 1,O)
4 2 ) 0(a),
g(0, 1)
L[G(O, I)],
9(0,0)
L(T3) 0(8)
where (a) is the 1-dimensional Lie algebra generated by 8.Recall that a subset 9’of a Lie algebra 29 is a subalgebra of 9 if ’3” is itself a Lie
2-5. THE LIE ALGEBRAS
"(0,
b)
37
algebra under the scalar multiplication, addition, and commutator bracket of 8.If 8 is a Lie algebra and 8, , g2are subalgebras of 9,then 8 is the direct sum of 8, and 8,,8 = 8,@ 8, , if considered as a vector space 8 is the direct sum of its subspaces 8,, g 2 ,and further, [a1, a2] = 0 for all a1 E gl , a, E 8,. Lemma 2.1
I
s ( a , b)
9(1,0) "(0, 1) s(0,O)
if a # 0, if a = 0, b # 0, if a = b = 0.
PROOF If a # 0, set $'+ = a-l$+, 3'- = a--l$-, f f 3 = - (ba-,/2) 8,8' = d. I n terms of the primed basis elements the commutation relations for %'(a,b) given by (2.17) become identical with those for 8(1,0). If a = 0, b # 0, set 8' = bd. I n terms of the basis f + , $-, f 3 , &", the isomorphism between 8 ( a , b) and 9(0, 1) is evident. This lemma shows that there are only three distinct Lie algebras of the form 8 ( a , b), up to isomorphism. However, it is often useful to consider the entire family { 8 ( a , b), a, b E @}of such Lie algebras because Eqs. (2.17) determine relationships between the nonisomorphic algebras %'(1, 0), 8(0, l), and 8(0,0). To spell out these relationships we introduce the notion of contraction of Lie algebras. Let 8 be an n-dimensional complex Lie algebra with basis {a,}, i = l,,.,, n. T h e structure constants Cfj of 8 relative to this basis are given by $3
[ai
,aj] =
n
1 Cfja, ,
i, j
= 1
,..., n.
k-1
If P = (P:)is a nonsingular n x n complex matrix we can define a new basis {a;}for 8 by a;= Pfa,, i = 1,..., n. I n terms of the new basis the structure constants become
x:=l
C::
P4P,hC;A(P-1):,
=
i,j , k
= 1,..., n.
l.A.r=l
We now introduce a one-parameter family P(t) of nonsingular matrices defined for all values of t > 0 in such a way that the matrix elements P$t) are continuous in t. Suppose this family of matrices has the property that the limits n
38
2. REPRESENTATIONS OF LIE ALGEBRAS
exist. It is easy to show that the {C$) are the structure constants of an n-dimensional Lie algebra (Saletan [l], Sharp [l]). If limi+,, P ( t ) = P where Po is a nonsingular matrix then the constants C:j” are structure constants for 9 relative to the basis
However, if limi-,o P ( t ) is a singular matrix or if the limit does not exist, the constants C:: may be structure constants for an n-dimensional Lie algebra 9’which is not isomorphic to 9.I n such a case we say that 9‘is a contraction of 99 (Wigner and Inonu [I], Saletan [I], Sharp [l]). According to Lemma 2.1 we can always find a basis for the Lie algebra 9 ( 1 , 0 ) in the form (2.17) where u # 0 and b is arbitrary. If u +0 in (2.17) we get in the limit either 9(0,1) or 9(0,0)depending on whether or not b = 0. This shows that 8(0,1) and S(0,O) are contractions of 9(1, 0). Similarly, 99(0,0) is a contraction of 9(0,1). This relationship between the Lie algebras S(a, b) will turn out to be of fundamental importance in special function theory. 2-6
Representations of %(a, b)
Let p be a representation of 9 ( u , b) on the complex vector space V and set
These linear operators obey the commutation relations
[J+, J-1
= 2a2J3- bE,
[J+, E]
=
[J3, J+]
[J-, El
=
J+,
= [J3, E] =
[J3, J-]
=
-J-,
(2.18)
0,
where now [ A , B] = AB - B A for linear operators A and B on V. Define the spectrum S of J 3 to be the set of eigenvalues of J3. T h e multiplicityof the eigenvalue X E S is the dimension of the eigenspace VA, V A= {W
E
V : J3v = Av}.
I n analogy with the example presented in Section 2-3 we will analyze the irreducible representations of 9 ( u , b) and for each such representa-
2-6. REPRESENTATIONS OF S(O,b)
39
tion find a basis for V consisting of eigenvectors of J3. T o be precise, we will classify all representations p of S(a, b) satisfying the conditions: (i) p is irreducible (ii) Each eigenvalue of J3 has multiplicity equal to one. There is a countable basis for V consisting of eigenvectors of J3.
(2.19)
These conditions are satisfied by the representations of L( 0,) classified in Section 2-3. Moreover, they enable us to construct representations of S(a, b) by mimicking the construction of representations of L(0,). T h e basic justification for the requirements (2.19) is that they quickly lead to connections between S ( u , b) and certain special functions. This claim will be thoroughly documented in the chapters to follow. Also, j 3 and 8 generate a Cartan subalgebra of S(a, b) (Jacobson [l]), so in analogy with the theory of finite-dimensional representations of semisimple Lie algebras it is natural to try to find basis vectors for V which are simultaneous eigenvectors of the operators corresponding to the elements of this Cartan subalgebra. (We will show that if p satisfies (2.19) then E = p(d) is a multiple of the identity operator and every non-zero vector in V is an eigenvector of E.) Condition (ii) can be written in forms which are apparently much weaker, though actually equivalent. In particular we could replace (ii) by (ii)'
J3
has an eigenvalue of finite multiplicity.
I t is not difficult to show that conditions (i) and (ii)' imply (ii). The necessary ingredients for the proof will be developed in the process of classifying the representations satisfying (i) and (ii), but the details will be left to the reader. We use condition (ii) for the sake of convenience. T o classify the representations of $(a, b) for arbitrary a, b E fZ' it is enough to consider the three cases: 9(1,0), Y(0, l), S(0,O). However, for the time being it will be convenient to treat these three cases simultaneously by studying $(a, b) without being specific as to the values of a and b. We will consider, therefore, a representation p of g ( a , b) on the complex vector space V such that (2.19) is satisfied. Our objective will be the enumeration of all possibilities for p. T o carry out this enumeration the following remarks will be helpful:
(A) Define the operator Ca,b on V by Ca,b = J+J-
+ a2J3J3- u2J3 - bJ3E.
(2.20)
2. REPRESENTATIONS OF LIE ALGEBRAS
40
It is easy to check that a E S(a, b). Thus, [ca,b
J’]
ca,bcommutes
== [ C a , b
, J-I
=
LCa,b
t
with every operator p(cy),
J31 = I C a , b EI t
= 0.
We will show that c o , b = hJ where I is the identity operator and A is a constant depending on p. ( B ) If S is the spectrum of J3, condition (ii) guarantees that S is countable and that there exists a basis for V consisting of vectors f m , J”f, = mf, , defined for every m E S . Suppose q E S . Then the equation [J3, J+]f, = J+f, leads to the result J3(J+f,) = ( q 1) J+fQ ; so either J+fQ = (,+,f,+l where c,+, is a nonzero constant and q 1 E S, or J+fQ = 0. Similarly, the equation [J3, J-If, = - J-f, implies either J-f, = qqfq-, where 7, is a nonzero constant and q - 1 E S , or J-f, = 0. T h e equation [E, J3] f , = 0 implies Ef, = p, f , for some constant pQ and [C,,b, J3] f, = 0 implies C,,,f, = A, f , for some constant A, . (C) Since p is irreducible the results of ( B ) show that the spectrum S must be connected, i.e., if q E S then S is of the form
+
S
=
+
+ n: n an integer such that nl < n < n2}
{q
where n, and n2 are integers. We do not exclude the possibilities n1 = - 00 or n2 = 00. In addition, if m, m 1 E S then gm+, , 7m+l # 0, since otherwise the irreducibility of p would be violated. 1 E S. Then the equation [E, J+]fm = 0 leads ( D ) Suppose m, m to (m+l(pm+l - pm) = 0, which implies p m = p,+l . Thus, pn, = p = constant for all m E S and E = pI is a multiple of the identity operator on V . A similar argument proves A, = A for all m E S so that Ca,b = AI. T h e equation ca,b ftn = Af, leads to the relations
+
+
+
+
for all m E S ,
5m7m u2m2 - u2m - 6mp = h
obtained from the definition (2.20)of ca,b, (If m - 1 $ S then 7, These relations can be rewritten in the form tmqm= h - u2m(m - 1)
Moreover, since [J+, J-] Ca,b
=
=
J-Jf
bpm.
=
0.)
(2.21)
2a2J3- bE the operator ca,bis also given by
+
dJ3J3
+ a2J3
-
6J3E- bE
From this expression we obtain the relations (2.22)
2-6. REPRESENTATIONS OF q ( a , b)
41
+
defined for all m E S, where [,n+, = 0 if m 1 6 S. Thus, Eq. (2.21) is valid for all m such that m E S or m - 1 E S. ( E ) If { y m , m E S } is a set of nonzero constants, we can introduce a new basis {fk} for V by means of the definition f,t,, = y m f m , all m E S. I n terms of the new basis: , ,
J-fk
= rlmfm--l
(2.23)
, Ym-1 tm = -t m , 7 ; Ym
Ym
= -"lm. Ym-1
+
(We assume f m + l = 0 if m 1 4 S and q m = 0 if m - 1 4 S.) Again the constants & , vi, must satisfy the condition
6'm 7'm
= X - a*m(m -
I)
+ bpm.
It follows from these considerations that for all m E S such that m - 1 E S we can choose the nonzero constants qm arbitrarily and define the constants tmby (2.21). (If m - 1 $ S then q, = 0 and tmis not defined.) We can always find a basis { fn, } of V such that q,,, , tmsatisfy (2.23) (without the primes). ( F ) T h e representation p of %(a, b) is uniquely determined by the constants A, p and the spectrum S of J3. The nonzero constants f , , 77, are not unique and may be chosen arbitrarily, subject only to conditions (2.21). This is as far as we can go without making any assumptions as to the values of a and b. At this point we use the fact that we need consider only the Lie algebras 9(1,0), 9(0,l), and 9(0,0). Theorem 2.1 Every representation of 9 ( 0 , 0 ) which satisfies (2.19) m,) and for which J+J- f 0 on V is isomorphic to a representation P(w, defined for p, w , m, E (Z' such that w # 0 and 0 Re m, < 1. S = {m, n: n an integer}. For each representation QP(w, m,) there is a basis for V consisting of vectors f , , m E S, such that
<
+
J Y m = mfm J+fm
=
9
ufm+l,
Co.ofm
E f m = Pfm J-fm
=
= (J+J-)fm =
The representations P(u,m,) and Qfi(
-w,
i
Wfm-1,
WYm
-
m,) are isomorphic.
(2.24)
42
2. REPRESENTATIONS
OF LIE ALGEBRAS
PROOF Set a = b = 0 in Eq. (2.22). If p is a representation of 9(0,0) satisfying (2.19) and such that J+J- + 0 on V, we have tmqm= X # 0 for all m such that m E S or m - 1 E S. Thus, there exists no element m, in S for which J+fm, = 0 for a nonzero fml E Vml and no element mp such that J-f,,,, = 0 for a nonzero fm, E Vma. If m, E S then S must be of the form {m, n: n an integer}. Without loss of generality we can assume m, is the element of S with smallest positive real part, i.e., 0 < Re m, < 1. Since tmqm= X # 0 we can also assume that 5m = qm = w for all m E S. Then X = w2 and w is determined only up to sign. This proves that every representation p satisfying the hypotheses of the theorem can be cast into the form (2.24). Conversely, it is easy to show that the representations Qp(w, m,) satisfy the hypotheses of the theorem. Q.E.D. The irreducible representations of 9 ( 0 , 0 ) for which J+J- = 0 are easily seen to be 1-dimensional. They are of little interest for our purposes so we will not bother to classify them.
+
Theorem 2.2 Every representation of 9(0, 1) satisfying (2.19) and for which E & 0 is isomorphic to a representation in the following list:
(i) T h e representations R(w, m, , p ) defined for all w , m, , p E C? such that p # 0, 0 Re m, < 1, and w + m, is not an integer. S = (m, n: n an integer). (ii) T h e representations tw,rdefined for all w , p E f2 such that p # 0. S = {-w n: n a nonnegative integer}.
<
+ +
For each of the cases (i) and (ii) there is a basis of V consisting of vectors fm defined for each m E S such that J Y m = mfm
= t+frn+lY
J'fm
=
Co.1fm
= pfm
Efm
9
= (m
J-fm
(J+J- - EJ3) f m
+
9
w)fm-19
= pwfm
(2.25)
-
(On the right-hand side of these equations we assumef, = 0 if m 6 S.) (iii) The representations Jw,r defined for all w , p E f2 such that p # 0. S = {-w - 1 - n: n a nonnegative integer}. For each of the representations there is a basis of V consisting of vectors f,,, defined for each m E S such that =
JYm J+fm
=
-(m
C0,lfm
mfm >
Efm
=
+w +
l)fm+l
9
(J+J-
- EJ')
fm
=
-pfm
9
J-fm
=
-pwfm.
pfm-1,
(2.26)
2-6. REPRESENTATIONS OF Y(o, b)
43
PROOF Let p be a representation of 9(0, 1) satisfying the hypotheses of the theorem and let S be the spectrum of p. Then Eq. (2.21) becomes t,r), = p(w m) for all m such that m E S or m - 1 E S. (We have set a = 0, b = 1, and h = pw in (2.21).) Let m, E S and suppose o + m, is not an integer. Then, S = {m, n: n an integer}. Without loss of generality m, can be taken to be the element of S with smallest positive real part, 0 Re m, < 1. According to the remarks in (E) there exists a basis for V such that 5 , = p, r), = w m for ail m E S. Thus, p is isomorphic to the representation R(w, m, , p ) listed in (i). If there is an element ml in S such that w m1is a nonnegative integer, the equation &,,r), = p(w m) implies - w E S and there exists a nonzero vector f - u E V such that J”f-w = -wf-,,, and J-f-u = 0. I n this case S = {-w n: n a nonnegative integer). There exists a basis of V such that , 5 = p, vrn= w m for all m E S. Thus p is isomorphic to the representation tu,rlisted in (ii). We say that this representation is bounded below. If there is an element m2 in S such that w m 2 is a negative integer we find - w - 1 €43 and - w $ S. Thus, S = { - w - 1 - n: n a nonnegative integer). There exists a basis of V such that 5 , = w m, 7 , = p for all m E S. In this case p is isomorphic to the representation JW,& listed in (iii). This representation is bounded above. Conversely, it is easy to verify that the representations (i)-(iii) actually satisfy conditions (2.19). Q.E.D. The irreducible representations of 9(0, 1) for which E = 0 can be considered as representations of 3T3 ; hence they are just the representations QO(w, m,) classified in Theorem 2.1.
+
+
<
+
+
+
+
+
+
+
Theorem 2.3 Every representation p of 9(1,0) satisfying conditions (2.19) is isomorphic to a representation in the following list:
(i) The representations D ( u , m,) defined for all complex p, u, m, such that m, u, m, - u are not integers and 0 Re m, < 1. S = {m, n: n an integer}. D ( u , m,) and D ( - u - 1, m,) are isomorphic.
+
+
<
(ii) The representations T;, p, u E C?, where 2u is not a nonnegative integer. S = {--u n: n a nonnegative integer}.
+
(iii) The representations 4; , p, u E C,where 2u is not a nonnegative integer. S = {u - n: n a nonnegative integer}. (iv) T h e representations Dr(2u) where 2u is a nonnegative integer. (24, u - 1,.“, -24 1, -u}.
s=
+
2. REPRESENTATIONS OF LIE ALGEBRAS
44
For each of these representations there is a basis of V consisting of vectors f, , defined for each m E S such that JYm
C,,,fm
= mfm
=
J-fm
=
(J'-
-(m
m f ' J
>
+ + J3J3
=
(m - u ) f m + l , Efm == /+fm
uIfm-19 - J3)fm
(2.27)
9
+ 1)fm
=~ ( u
*
(We make the convention on the right-hand side of Eqs. (2.27) that = 0 if m I$ S.)
fm
PROOF Let p be a representation of 9(1,0) satisfying the hypotheses of the theorem. From (2.21) we obtain the equations t,r), = u(u + 1) - m(m - 1) = - ( m u)(m - u - 1) valid for all m such 1). Suppose there that m E S or m - 1 E S. Here, we have set h = u(u is an element m, in S such that neither m, u nor m, - u is an integer. I n this case the product t,~,can never be zero for any element m or m - 1 in S. Thus, S = {m, + n: n an integer}. Without loss of generality we can assume that m, is the element in S with smallest nonnegative real part: 0 Re m, < 1. According to remark (E) there exists a basis of V such that 5, = m - u - 1, r), .= - m - u. Thus, p is isomorphic D ( - u - 1, m,) to the representation B ( u , m,) listed in (i). D ( u , m,) since both u = u, and u = -u, - 1 lead to the same value of A. If the spectrum of p takes the form S = {ml n, n a nonnegative integer}, the formula tmvm = h - m(m - I), m E S , implies h = ml(ml - l).Settingm, = -uweobtainh = u(u 1)andS = {-u n: n a nonnegative integer}. As before we can find a basis of V such that 4, = m - u - 1, qnl = -m - u . Here, p is isomorphic to the representation 1;. T h e 1: is bounded below. If the spectrum of p takes the form S = {m, - n: n a nonnegative integer}, the equations (,,r), = h - m(m - 1) for m E S or m - 1 E S, imply X = (m, I)m,, i.e., t,,+l = 0. Setting m2 = u we have h = u(u 1) and S = {u - n: n a nonnegative integer}. Choosing tnL= m - u - 1, T ) , ~= -m - u we see that p is isomorphic to 1:
+
+
+
<
+ +
+
+
+
(bounded above).
+
If S contains finite elements m3, mq such that S = {m3, m3 1,..., m3 k ,..., m4} the equations .f,,,r),), = X - m(m 1) imply h = m3(m3 - 1) = (m4 l)m4 (since = qn,, = 0). Set mq = u and m 3 = u - n where n is a nonnegative integer. Thus, u(u 1) = ( u - n)(u - n - I). T h e only possible solution of this equation is n = 224. We conclude that S = { - u , - u + 1,..., + u} and p is isomorphic to the representation D(2u). Since the possibilities for S have been exhausted, the theorem lists all representations p satisfying (2.19). Q.E.D.
+
+
+
+
2-7. REALIZATIONS OF %(a, b) IN TWO VARIABLES
2-7
Realizations of
"(0,
45
b) in Two Variables
I n accordance with our general program we will try to find realizations of the irreducible representations p of " ( a , b) listed in Section 2-6, such that V becomes a vector space of analytic functions and the operators p(a), a E " ( a , b), form a Lie algebra of analytic differential operators acting on V . As a first step we will determine some possible candidates P(4-
Suppose, first, that the p(a) are differentia! operators acting on a space of analytic functions of two complex variables, x and y . For the moment we will not be concerned with the precise domains of the functions in this space. Define the differential operators J+, J-, J3, E by
AB+)= J+,
P(B-)= J-,
Since p is a representation of usual commutation relations [J+, J-]
= 2a2J3- bE,
p(f3> =
J3,
~ ( 6=) E.
s(a,b) these operators must satisfy the
[J3, J*]
=
& J*,
[J*, El
=
[J3, El
= 0.
(2.28)
T h e number of possible solutions of Eqs. (2.28) is tremendous. To obtain useful results it is convenient to make more restrictive assumptions as to the form of the operators p(0r). Thus, we assume that these operators take the form
a
(2.29)
J3=37,
where p is a complex constant and k, j are functions of x to be determined. T h e operators (2.29) are natural generalizations of the angular momentum operators (2.13) related to 0, . However, we will not give a detailed justification for this choice of differential operators now. In Chapter 8 a theory of generalized Lie derivatives will be developed which will enable us to classify all solutions of (2.28) by differential operators in two variables. It will follow from the classification that nothing of importance for special function theory is lost through the restrictions
(2.29).
T h e operators (2.29) automatically satisfy all of the commutation relations (2.28) except [J+, J-] = 2a2J3- bE. An easy computation shows that this last relation will be satisfied provided k a n d j are solutions of the differential equations dk (x) dx
+ k(x)2 = -a2,
dx
(x)
+ k(x)j(X) = - bP 2 '
(2.30)
2. REPRESENTATIONS
46
OF LIE ALGEBRAS
Thus, to find all operators of the form (2.29) it is sufficient to find all solutions of Eqs. (2.30). 8(1,0)
If a = 1, b = 0, Eqs. (2.30) have the solutions type A
or type B
j(x)
k ( x ) = i,
=
p-aZ
where p , q are complex constants and i = d q .T h e type designation is based on the classification of factorization types due to Infeld and Hull [l]. 8 ( 0 , 1)
If a = 0,b k(x)
=
1, the solutions are type C'
1 x+p'
j ( x ) = - tL(x
=-
+P)
4
4 +
x+p
or type D'
P
k ( x ) = 0,
j(x) = -2
+q
where p and q are complex constants. Q(0,O)
If
a =
b = 0, the solutions are type C" 1 K(x) = x
or type D"
+p'
k ( x ) = 0,
4 =x+p
j(x) =q
where p and q are complex constants. Suppose we are able to realize an irreducible representation p of S ( a , 6 ) classified in Section 2-6 in such a way that the operators p ( m ) are
differential operators of one of the types listed above and the basis space V is a space of analytic functions of x and y . T h e basis functionsf,(x, y ) , m E S, satisfy the equations
a
J?fm(X,.Y) == --fm(x,
aY
Y ) = m-frn(x, Y ) .
Thus, f m ( x ,y ) = gnl(x)emg for all m E S , where g,(x) is an analytic function of x. Since p is irreducible, we have C 0 . b fm
= hfm
m E S,
(2.31)
2-7. REALIZATIONS OF %(a,b) IN TWO VARIABLES
47
where the constant X is uniquely determined by p. However, in terms of our realization Cm,*is a second order partial differential operator and the relations (2.31) are a system of second order partial differential equations for the basis functionsf,(x, y). Sincef,,(x, y ) = g,(x)emv the dependence on y can be factored out and the relations simplify to a system of second order ordinary differential equations for the functions g,(x). The g,(x) will turn out to be special functions and the action of p will yield recursion relations, generating functions and addition theorems for these functions. T o determine the possible functions g,(x) which can arise in this way we recall Ca.b = J'J+ u2J3J3- u2J3- bJ3E and evaluate (2.31) for each of the operator types. The following differential equations are obtained for g,(x). For type A operators Eq. (2.31) becomes
S(l,O)
According to Theorem 2.3, X
= u(u
+ 1). One solution of this equation is
+ w)-"F(m - u, -q
grn(x)= w(m-Q)/2(1
+
- u; m - q
+ 1; - w )
where w = tan2[(x p)/2]. If m - q is not a positive integer there is a linearly independent solution of the form g,(x)
+ w)-"F(q - u, -m
= w'q-m)-/2(1
- u; q - m
+ 1; -w).
These results show the connection between type A operator realizations of 9(1,O) and the hypergeometric functions. For type B operators we obtain -e-iz
d dx
. -gm( d dx
]
x) t [ -q 2e-2iz
+ 2mqe-'"]
g,(X)
+ 1) grn(x)
=~ ( u
where u(u + 1) = A. If 2u is not an integer this equation has the linearly independent solutions g,(x)
=
(24z)u+le-*" ,F,(u - m
g,(x)
=
(2q2)-"
+ 1; 2u + 2; 2qz)
and e-9"
,F1(-u
-
m ; -224; 2qz)
2. REPRESENTATIONS OF LIE ALGEBRAS
48
where z = -ierix. This shows the relation between type B operator representations of 8(1,O) and the confluent hypergeometric functions. 8 ( 0 , 1)
For type C’ operators Eq. (2.31) becomes
= pwgm(x),
, I= p w .
If m - q is not an integer this equation has the linearly independent solutions
.26
+ +
+ 1; x2/4)
+
where t = ( m - q)/2 and 77 = ( p / 2 ) ( m q 2w 1). These solutions are “functions of the paraboloid of revolution,” and are closely related to the confluent hypergeometric functions. Type D’ operators yield the equation , I= pw.
This is the parabolic cylinder equation which has as linearly independent solutions the parabolic cylinder functions Dmtw(2/@), Dm+w(- &&). Ce(0,O) sakfy
The functions g,(x)
corresponding to type C” operators
This is Bessel’s equation and its solutions are cylindrical functions. In particular, the Bessel functions JnL(wx)and J-,(wx) satisfy this equation. For type D“ operators we obtain d2 -dxz g r n ( X )
= wzgm(x),
, I=
~~a
T h e solutions e i i & are special functions of a very simple kind. This brief survey shows the deep relationship between functions of hypergeometric type and the Lie algebras %(a,b). I n the next three chapters this relationship will be examined in detail.
2-8. REALIZATIONS
2-8
OF
'3(a,b) IN O N E VARIABLE
49
Realizations of 9 ( a , b ) in One Variable
I n the last section we found differential operators J+, J-, complex variables, satisfying the commutation relations:
[J3, J']
=
J+,
J3,
E in two
[J+, J-] = 2a2J3- bE, [J3, I-] = -J-, (2.32) [J3, El ='[I+, El = [J-, El = 0.
An analogous problem, which will be solved now, is to find realizations of (2.32) in terms of linear differential operator's in one complex variable z. I n particular we look for all nonzero differential operators of the form d J3=h+z-,
E=p
dz
J'
= ji(z) +j2(z)
d
J-
9
= h(z)
+
k2(z)
(2.33)
d
such that the commutation relations (2.32) are satisfied. Here, X and p are complex constants and j , ,j2, K, , k2 are functions of z to be determined. J3 has been chosen so that its eigenfunctionsf,n will be of the form z ~ - i.e., ~ , powers of z. Although this choice of J3 appears special it will be shown in Chapter 8 that every realization of 9 ( a , b) by differential operators in one complex variable is equivalent to a realization of the form (2.33). T h e operators (2.33) satisfy the commutation relations (2.32) if and only if c3 jl(z) = clz, j z ( z ) = czz2, k,(z) = k d z ) = c4 2 '
where the constants c, c2c4' =
,..., c4 satisfy the equations 4 2 ,
czcg
+
ClC4 =
-2a *A
+ bp.
(2.34)
I t is enough to solve these equations for the three cases 9(1, 0), 9(0,I), q o , 0).
%(I, 0 ) Conditions (2.34) become
~
2 = ~ -1, 4
~
2
By a change of variable z if necessary, we can assume
+
~ tic4 3 =
c2 = -c4
-2. =
1.
This leaves only the condition c, - c, = -2. Thus, the general solution is d J3=X+z--, E=p dz (2.35) d J+ = (2h + c3)z + 2 2 J- = 5 dz ' z dz where t3is a constant.
d
50
2. REPRESENTATIONS OF LIE ALGEBRAS
+
~ p. 4 We 9 ( 0 , 1) Conditions (2.34) become c,c4 = 0, ~ 2 ~ ~3 1 = assume p # 0. If c, = 0 then c4 # 0 and clc4 = p. By a change of the variable z if necessary, we can assume c1 = p, c4 = 1. Thus, we obtain the solution d J3=h+z-, E=p, dz (2.36) J+=pz, J-=%+C 3 € @. z dz'
If, however, c4 = 0 then c, # 0 and c2c3 = p. We can assume c, = p, c, = 1 to obtain the solution d J3=h+z-
dz '
J+ = clz
d +22 dz '
E=p, J - = I". z'
c, E
c.
(2.37)
+
3 = 0. In order 8(0,0) Conditions (2.34) are c2c4 = 0, ~ 2 ~ clcp that Jf $ 0, J0 we must have c, = c4 = 0. By a change of the variable z if necessary, we can assume c1 = c, # 0 and obtain the solution d J3=h+z--, E=p, J+=clz, J-= 5 C,E @. (2.38) dz 2 '
+
In the next three chapters the differential operators (2.35)-(2.38) will be used to construct realizations of the abstract representations of 3 ' ( a , b) classified in Section 2-6.
CHAPTER
3 Lie Theory and Bessel Functions
The machinery constructed in Chapters 1 and 2 will here be applied to the study of Bessel functions. As was demonstrated in Section 2-7 these functions are related to the representation theory of 9(0,0). The Bessel functions appear in two distinct ways: as matrix elements of local irreducible representations of G(0, 0) and as basis functions for irreducible representations of 9(0,0). The first relationship will yield addition theorems, and the second will yield generating functions and recursion relations for Bessel functions. @ (&), the nontrivial part of the representation Since 9 ( 0 , 0 ) 9, For a theory of theory of "(0,O) is concerned with the subalgebra 9,. Bessel functions, it is sufficient to study the representation theory of F3 and the local Lie group T , . Thus, we restrict ourselves to Y, throughout this chapter. The Euclidean group in the plane E, is a real 3-parameter global Lie group whose Lie algebra is a real form of Y 3 The . faithful irreducible unitary representations of E, are well known as is the fact that with respect to a suitable basis, the matrix elements of these representations are proportional to Bessel functions of integral order (Vilenkin [I], Wigner [2]). In Section 3-4 we will study the relationship between local representations of T , and unitary representations of E, . The computations involved in this chapter are rather simple and will serve as an introduction to the much more complicated theory of hypergeometric and confluent hypergeometric functions.
3. LIE THEORY AND BESSEL FUNCTIONS
52
3-1
The Representations Q(w. m,)
Each irreducible representation of 9(0,0) classified in Theorem 2.1 is designated by the symbol Qp(w, m,) where p, w , m, are complex constants such that w # 0 and 0 Re m, < 1. T h e spectrum S of this representation is the set (m, n: n an integer} and the representation space I'has a basis {f,: m E S},such that
<
+
JYm
= mfm
= wfm-1,
J-fm
T h e operators J+, J-,
[J3, J']
=
Efm
9
J3,
f J*,
= pfm
Co,"fm
J'jm
9
= wfm+l*
= (J+J-)fm
w"f,
1
(3.1)
-
E satisfy the commutation relations
[J+. J-I
[J*, E]
= 0,
=
[J3, E]
= 0.
We will try to find a realization of the algebraic representationQr(w, m,) in terms of linear differential operators acting on a vector space of functions of one complex variable. I n particular we will realize Qp(w, m,) in such a way that the differential operators take the form (2.38). T h IS ' can be accomplished as follows: Let Yl be the complex vector space consisting of all finite linear combinations of the functions h,(z) = zn,n = 0, f l , &2,... . In Eqs. (2.38) set h = m,, c1 = w , i.e., J+
Define the basis vectorsf, of Yl byf,(z) and n runs over the integers. Then, (m,
= WZ, =
2
h,(z) where m
=
m,
+n
+ z dz ") zn = (no+ n)zn = mfm ,
J3fm
=
J+fm
= (WZ)Z" = W Z " + ~ = wfm+l>
J-fm
= (w/z)z" = wzn-l - wfm-1
Efm
J-
(3.3)
= pfm
for all m E S. These equations coincide with (3.1) and yield a realization of Q p ( w , mo), Now that we have realized the abstract representation Q*(w, m,) in terms of differential operators acting on analytic functions, we can apply the Lie theory of transformation groups to obtain a multiplier representaHowever, before tion of the local Lie group whose Lie algebra is 9(0,0). beginning this computation it will be convenient to simplify the problem
3-1. THE REPRESENTATIONS Q(w, m,)
53
slightly. Since 9(0,0) 9, 0(a)and p ( 8 ) = E is a multiple of the identity operator for every irreducible representation p, the nontrivial part of the representation theory of 9 ( 0 , 0 ) is given by the action of p on 9, . We will lose nothing of importance for special function theory if we study only those representations Qp(w, m,) for which E = p = 0. Moreover, the representations Qo(w, m,) induce an irreducible represenT h e action of tation Q(u, m,) of the 3-dimensional subalgebra F3. Q(u, m,) is obtained explicitly from Eqs. (3.1)-(3.3) by suppressing the operator E. I t was shown in Section 1-2 that F , is the Lie algebra of the local Lie group T 3 ,a multiplicative matrix group with elements
We will extend the representation Q ( w , m,) of F , , defined on Vl , to a multiplier representation of T , . Let a, be the vector space of all functions of z analytic in some neighborhood of the point z = 1. Clearly V1C a1. According to Theorem 1.10 the differential operators (3.5)
generate a Lie algebra which is the algebra of generalized Lie derivatives of a multiplier representation A of T, acting on a, . We will verify this fact and compute the multiplier v. T h e action of the group element on , T E @, is obtained by solving the differential equations exp ~j~ dz
_ -- z, dT
d
- v ( z 0 , exp T 9 ' ) = mov(zo,exp T f 3 ) dT
with initial conditions z(0) = 9,~ ( e) 9, = 1. (See Section 1-2 for the definition of exp T j ' . ) T h e solution of these equations is Z(T)
exp T f 3 ) = e m o r ,
= zoer,
v(z0,
Thus, i f f € 67, is analytic in a neighborhood of 9 then [A(exp T f 3 ) ) f ] ( Z o ) for 1
T
=
[exp r J 3 ] f ( z 0= ) emo'f(zOef)
1 sufficiently small. Similar computations yield [A(exp b Y + ) f l ( z " )= exp(bwF)f(zO), [A(exp c f - ) f ] ( z o )= exp(cw/zo)f(zo).
54
3. LIE THEORY AND BESSEL FUNCTIONS
If g E T, is given by (3.4) we find g
= (exp b$+)(exp
Hence, for 1 T 1, 1 b 1, 1 f E 152,takes the form
c
cf-)(exp
~$3).
1 sufficiently small the operator A(g) acting on
~ ~ w i (=z [WXP )
c f - 1 WXP
V+)WXP
T ~ 3 m z )
= exp(bwz)[~(expCY-){A(~XP 4 3 ) m z )
= exp w (bz
+ f) [A(exp . f 3 ) ) f l ( z )
= exp ( wbz
+ -z + mo7)f(eTz), WC
(3.6)
defir.ed for z in a sufficiently small neighborhood of z = 1. The m*dtiplier v is given by v(z,g) = exp(wbz + wc/z m,,.). Conversely, (3.6) defines a multiplier representation of T, whose generalized Lie derivatives are the differential operators (3.5). The elements of Vlare analytic for all values of z # 0, so iff E V, and z # 0 the expression
+
[A(g)f](z)
= exp ( wbz
+ “c + moT)f(eTz) 2
defines an element of C?l1 for all g E T, , not just for g in a sufficiently small neighborhood of the identity. We define the vector subspace 7, of a, (the completion of TI, see Section 2-2) as the space of all finite linear combinations of functions of the form A(g)f for all g E T,, f E Vl . By definition, TIis invariant under A: Ak)feTl
for all
f € v 1 , g e T,.
Furthermore, iff E Y1, g, , g, E T, , then
A k d f = AkI”k2)fl.
(3.7)
a,
By restricting from to Q, the space on which the local multiplier representation A acts, we have been able to extend A over the whole , has been “exponentgroup T, . Thus, the representation Q(w, m,) of F iated” to yield a global representation of T, . The basis functions fm(z) = h,(z) = zn, n = 0, & l , ..., m = m, n, for V,form an analytic basis for 9,(see Section 2-2). This is true because every element f in Vlis analytic at all points z # 0 in G’; hence, f has a unique Laurent expansion
+
c W
f(2) =
n=--m
unzn,
I%,
3-1. THE REPRESENTATIONS Q(w, m,)
55
which converges for all z # 0. Thus, we can define the matrix elements A,,(g) of the operators A ( g ) on 4, by
Using (3.6) we obtain exp wbz (
+ -z + (k + m , ) ~ ) WC
=
f
l=-m
&(g)z’
(3.9)
which can be regarded as a generating function for the matrix elements. From A(g,g,)h, = A(g,)[A(g,)h,] there follows the identity
Comparing coefficients of h,(z) = zl we obtain the addition theorem
c A,,(gl) m
Az,(g,g,)
=
j=-m
Ajk(g2),
I, k- = 0, 31, f2,...9
(3.10)
valid for all g, ,g, E T , . T o find an explicit expression for the matrix elements A,,(g) it is sufficient to compute the coefficient of zi on the left-hand side of (3.9). Thus,
=
e(k+mo)‘
m
(Wb)s(CU)s+k--l
1”
s!(s
I, k
& l , &2 ,..., (3.11)
l=--m
+ k - I)!
’
so Azk(g)= e(k+mJT(c
w)k-z
F
(w2bc)S s!(s
+ k - l)!,
= 0,
56
3. LIE THEORY AND BESSEL FUNCTIONS
where the sum extends over all nonnegative integral values of s such that the summand is defined. Specifically,
where the function F , , is defined by its power series expansion (A. 21). These two cases can be combined into the single expression
valid for all integral values of I, k. If bc # 0 the matrix elements are closely related to Bessel functions. To see this we introduce new group parameters r, v in place of b, c by
<
< arg b, arg c T,or briefly, r = (ibc)1/2and v = (b/ic)1/2. where r-, With these definitions it follows that b = rvi2, c = -rv-1/2, and this coordinate change is one-to-one if bc # 0. I n terms of the coordinates T , ZI the matrix elements (3.11) can be expressed as A,,@) = e(m,,+k)T(
-q~)l-k
J~-~(-uY),
k = 0,f l , &2 ,...
(3.14)
[see (A. 2O)J. T h e functions J l are Bessel functions of integral order. I n particular, for T = 0, ZI = -1, w = -1, k = 0, substitution of (3.14) into (3.9) yields the well-known generating function exp
(z
-
z-l)] =
f
~~(r)zl.
(3.15)
k-CC
for Bessel functions. We can obtain addition theorems for the functions di; by substituting (3.12) into (3.10). After some simplification one finds
3-2. RECURSION RELATIONS FOR MATRIX ELEMENTS
57
where n is a nonnegative integer and b, , b, , c1 , c2 E @. Some special cases of this formula are of interest. If b, = 1, b, = c1 = 0, c, = x, (3.16) reduces to
which is the power series expansion for .(We have used the fact that p,(K;0) = 1.) If b, = 1, b, = 0, c, = x, c, = y, then
0, c1 = -c2
If 6 , = x, b,
=
For b, = c1
= x,
=
1 there follows
b, = c, = y we obtain
.O F d I i I
+ 1; x2)
$1(1
n -j
I
+1;~~).
T h e addition theorem (3.10) can also be used to derive identities relating Bessel functions of integral order. However, it will be more convenient to derive these results in Section 3-3 as special cases of identities relating Bessel functions of arbitrary order. 3-2
Recursion Relations for t h e M a t r i x Elements
T h e matrix elements A,,(g), g E T, , computed in (3.1 1) are entire analytic functions of the group parameters T , b, c. Hence they can be considered as analytic functions on the group manifold. Denoting by a(T,) the space of all functions on T, which are analytic in some neighborhood of the identity element, we see that A,, E a(T 3 )for all integers I, k, where the Aj, are matrix elements corresponding to the representation Q(W, mo).There is a natural action P of T3 on a(T,) as a local transformation group. P is defined by
58
3. LIE THEORY AND BESSEL FUNCTIONS
for all j~ @(T3)and all g', g in a sufficiently small neighborhood of E T3 (the neighborhood depends on f). From (3.17) it is evident that
e
P(glg2)f = Pkl)[Pk2)fl,
if g, ,g, , and g, g, are in the domain off. Thus, P defines T3as an effective local transformation group on the 3-parameter group manifold of T3 . The action of P on the matrix elements A j k is easily determined:
for all g, g' E T3 ,j , k = 0, f1, &2,... . Comparing this expression with (3.8) we see that for fixed j the functions { A j k , k = 0,fl, &2,...}, form a basis for the representation Q(u, m,) of T , . The Lie derivatives J+, J-, J3 defined by
for all f~ @( T3),must, therefore, satisfy the commutation relations
and act on the basis vectors Ajk as follows:
These expressions give recursion relations and differential equations for the matrix elements Ajk. T o evaluate them it is necessary to compute the Lie derivatives defined by (3.19). The. elements of O!(T,) can be considered as analytic functions of the group coordinates b, c, 7 , so the Lie derivatives are linear differential operators in these three complex variables. However, if we wish to relate our results to Bessel functions it is more convenient to use the group coordinates Y , c,7 , (3.13). As we
3-3. REALIZATIONS OF Q(w, m,) IN TWO VARIABLES
59
have seen, in terms of the I, v coordinate system the matrix elements are given by (3.14): A&)
= e(mo+k)r(-v)l--k
Jl--k(-W+
This coordinate system is not defined over the whole group but only for those group elements such that bc # 0. If g has coordinates (I, v , r ) and
1
g' = (0
0
0
e-7'
0
0 0
e7' 0
0 0
")
1
a simple computation shows that the coordinates of gg' are given by e-lc'v
(r[l - 2-
Y
2erb' + tp,- 4
7 1 b'c'
1
lI2
v
+ 2erb'/rw
[ 1 - 2e-7cIv/r
for I b' I, I c' I sufficiently small. From this result and the definition (3.19) of the Lie derivatives there follows
Substituting (3.14)and (3.21) into (3.20) we obtain, after factoring out the dependence on v and r ,
where n is an integer.
3-3
Realizations of Q(w, m,) in Two Variables
In the previous sections realizations of the representation Q(w, m,) of .Z3have been determined on spaces of functions of one and three complex
variables, respectively. We will now find realizations of this representation on spaces of functions of two complex variables, x and y. In particular? we look for functionsf,,(x, y) = Z,(x) emu, such that
60
3. LIE THEORY AND BESSEL FUNCTIONS
+ k: k an integer), where the differential operators
for all m E S = {m, J*, J" are given by
(3.24)
These operators are the type C" operators classified in Section 2-7. (The constants p , q occurring in the expression for the type C" operators can be set equal to 0 without any loss of generality.) The content of (3.23) is, thus, a series of equations relating the functions Z,(x):
The complex constant o in these equations is clearly nonessential; we could remove it by making the change of variable x' = w x . Hence, we wilI assume w = -1. (For this choice of w , Eqs. (3.25) agree with the conventional recursion relations for cylindrical functions, (A. 23).) Nonzero solutions of these equations are well known. If m, = 0 so that the elements of S are integers, we see from (3.22) that Z,(x) = J,,,(x) is a solution for all m E S . Furthermore, for arbitrary complex values of m, we can obtain the solutions Znt(x)= J m ( x ) , J-,,,(x), H:)(x), HZ'(x), Nm(x),for all m E S (Magnus et al. [l]). Thus, the Bessel functions of first and second kind, the Hankel functions of first and second kind, and the Neuman functions each satisfy the recursion relations (3.25) for o = - 1. Each of these functions is analytic for all values of x except x =
0.
If, conversely, functions Z,(x) defined for each m E S satisfy (3.25) for w = -1, then the vectors fin(x, y ) = Z,(x) emu form a basis for a . This Lie algebra realization of the representation Q(- 1, m,) of 9, representation induces a local representation of T,. In fact if 02 is the space of all functions analytic in some neighborhood of the point (9, y") = (1, 0), the Lie derivatives
define a local multiplier representation T of T3on 0. From Theorem 1.10, [T(exp b f + ) f ] ( x , eY) = f ( x ( b ) , ey(*)),
b E C?,
j c @,
3-3. REALIZATIONS OF Q ( u , ~ , ) IN TWO VARIABLES
61
where dx
db (b) = e*(b),
-dY( b )
=
db
-
eu(b)
x(b) ’
x(0) = x, y ( 0 ) = y .
T h e solution of these equations is
where for convenience we use the new variable t = eu. Similar computations yield
g =
1 ‘j, 0
0 e;
eT o
0
c
0
0
1
a g in a then g = (exp b$+)(exp c2-)(exp T f ’ ) . So, for f ~ and sufficiently small neighborhood of the identity we have
An explicit computation gives
[T(g)f](x, t )
=
1
+-
(3.26)
defined for I 2bt/x 1 < 1, j 2c/xt I < 1. According to Section 2-2, our on the space generated realization of the representation Q(- 1, m,) of F3 by the functions’f,,(x, y ) = Z,(x) emu, m E S, can be extended to a local representation of T3 where the group action is given by (3.26). In particular, the functions fn(x, y ) form an analytic basis for this space, as can easily be seen from (2.2) and the y-dependence emu of the basis vectors. Thus, the matrix elements of this local representation
62
3. LIE THEORY AND BESSEL FUNCTIONS
with respect to the basis f, are uniquely determined by Q(-I, m,) and without computation we immediately obtain the relations
(3.28)
where the matrix elements A,,(g) are given by (3.12) (o = -1). T h e region of convergence of these relations is determined by examining the singularities of the functions on the left-hand side of (3.28). Since 2, , m E @, is analytic in x for all nonzero values of x, it follows that the infinite series (3.28) converges absolutely for I 2bt/x I < 1; I 2c/xt I < 1. Explicitly,
(3.29)
Several of the fundamental identities for cylindrical functions are special cases of this formula. If c = 0, t = 1, Eq. (3.29) becomes
If b = 0, t = 1, one obtains
I n the case where 2, of Lommel.
= J , , (3.30) and (3.31) are known as the formulas
3-4. WEISNER'S METHOD FOR BESSEL FUNCTIONS
63
If bc # 0 we can introduce the coordinates Y , o defined by (3.13), such that b = rv/2, c = - ~ / 2 v . I n this case A,,(g) = e(mo+K)T(
-v)t-k
Jl-k)
and Eq. (3.29) simplifies to
m
1x1
T
Izl
(3.32)
For 2, = Jm this is a generalization of Graf's addition theorem (ErdClyi al. [l], Vol. 11, p. 44).
et
3-4 Weisner's Method for Bessel Functions Expressions (3.27) are valid only for group elements g in a sufficiently small neighborhood of the identity element of T 3 . However, we can also use the type C" operators (3.24) to derive identities for cylindrical functions associated with group elements bounded away from the identity. T h e following remarks are pertinent. If f ( x , t) is a solution of the equation Cosf = coy, i.e., (3.33)
then the function T(g)f given formally by
satisfies the equation Co,o(T(g)f) = w2(T(g)f) whenever T(g)f can be defined. This follows from the fact that Co,ocommutes with the differential operators J+, J-, 53. Furthermore, iff is a solution of the equation (xiJ'
+ 4-+ x3JS)f(x, t )
Mx,t)
(3.34)
for constants x1 , x2 , x3 , A, then T(g)f is a solution of the equation
+ x2J- + x3J3)T(g-')I[T(g)fl
[T(g)(xiJ+
= V'"g)fI
3. LIE THEORY AND BESSEL FUNCTIONS
64
where
+ xzJ- + x3J3)T(g-')
+ (xze-. + cx3)J- + x3J3.
= (xleT - 6x3)J+
T(g)(x,J+
(3.35)
This is a consequence of Eq. (1.43). As an example of the application of these remarks consider the function f ( x , 1 ) = J,,,(x)t", m E @. Here Co,,f = f, = mf, so the function
JY
satisfies the equations
If
7
=
b
=
0, c
=
-1, we can write (3.36) in the form
h(x, t ) = (x2
2x
+ t) Im -ml2
[(XZ
+
y]+ (2
xtp.
Since x-mJ,,,(x) is an entire function of x, h has a Laurent expansion about t = 0:
c hn(x)t", m
h(x, t )
=
n=-m
1 x t I < 2.
Substituting this expansion in the first equation (3.37) we see that h,(x) is a solution of Bessel's equation
for each integer n. T h e function h(x, t ) is bounded for x = 0, so we must have h,(x) = c,J,(x), c, E C?' [see (A.22)]. Thus,
From the second equation (3.37)' (- J- + J3) h ( x , t ) = mh(x, t ) , it follows that cnfl = (rn - n ) c , . T h e constant co can be determined
3-5. THE REAL EUCLIDEAN GROUP
€3
directly by setting x = 0 in h(x, t ) . T h e result is co = l / r ( m whence c, = l / r ( m - n 1). Thus, we obtain the identity
+
I x t I < 2.
65
+ I),
(3.38)
Equation (3.38) is obviously not a special case of (3.29). For other examples of generating functions derived by this method see Weisner [3]. T h e above results were all obtained by using type C" operators. T h e type D" operators lead to no new results for special functions so we will not consider them here. 3-5
The Real Euclidean Group E,
T h e Euclidean group in the plane, E,, can be defined as the real 3-parameter group of matrices cos0 g(0, xl, x p ) = sin 0
( 0
-sin 0 x1 cos 0
0
(3.39) 1
where x1 , xg are real and 8 is a real variable determined up to a multiple of 277. As is well known, E, acts as a transformation group in the plane. If y = (yl ,yz)is a point in the plane R2the action of E, is given by y
---f
+ ( y z - xz) sin 0, sin 0 + ( y z x2) cos 0).
g-ly = ((yl - xl) cos 0 - (yl
-
xl)
-
This action corresponds to a translation (yl ,y 2 )-+ (yl - x1 ,y z - xz), followed by a rotation through the angle 8 in a clockwise direction about the point (0, 0). I t is easily verified in terms of the group parameters that the group multiplication is given by g(4 x1 , = g(O
R(O',
.;x;) 1
+ O', x; cos 0 - x;
sin 0
+ x1 ,x; sin 0 + x; cos 0 + xz).
(3.40)
T h e identity element of E, is the identity matrix g(0, 0, 0) and the inverse of g(8, x1 , x2) is g-1(0,
, x2) = g( -0,
-xl
cos 0 - x2 sin
0, x1 sin 0 - x2 cos O),
66
3. LIE THEORY AND BESSEL FUNCTIONS
since g(8, X, ,x2)g( -8, -xl
cos 8 - x2 sin 8, x1 sin 8 - x2 cos 8) = g(0, 0,O).
As a basis for the real Lie algebra 8, of E, we choose the elements fl
=
i),
f
[),
f
f 2 =
:)
(h 8 0 -1
J%=
0
(3.41)
with commutation relations [A1 9
A1 = 0,
[ A* A1 = f
2
9
[ A A1 = - f 1 + iYl,2-= y 2+ 9
matrices ,$+ = - f 2 Y 3= if3, i = satisfy the commutation relations
The
complex
[f3,
$+I
47, =
f-1
[f3,
%+I
=
-f-,
f-I
[ff,
=0
which are identical with the relations (1.41) for a basis of 9,Thus, . the complex Lie algebra generated by the basis elements (3.41) is 9,We . is the complexification of 8,and 8,is a real form of F , say that 9, (See Helgason [l], p. 152). Due to this relationship between F , and 8,, induces an irreducthe abstract irreducible representation Q(w, m,) of ible representation of 8,. There is another matrix realization of E, which we will find useful. Namely, we define matrices [eie
x1 ,x2 E R,
(x2 - ixJ2 \ (-xz - ix1)/2
0
0 <8
< 2a,
1
I
,
mod 27~.
(3.42)
These matrices form a realization of E, isomorphic to the realization by the matrices (3.39). I n fact,
g(4
.; +
x1 9 x2)g(B', = g(8
7
B', x; cos 8 - x; sin 8
+ x1 , x; sin 8 +
X;
cos 8
+ x2)
in agreement with (3.40). I t will sometimes be convenient to use, instead of the coordinates xl, x 2 , the polar coordinates r 2 0 and defined by x,
+ ix, = r&,
3-6. UNITARY REPRESENTATIONS OF LIE GROUPS
67
where we assume that r e 9 = 0 implies Y = v = 0. In terms of these new coordinates the matrices (3.42) take the form
1
If g' has coordinates [O', Y ' , v'] and g" has coordinates [O", Y", q"] then an easy computation shows g'g'' has coordinates [O, Y, v] where 0 = 8' + O",
reiq
= y'efo'
In particular, the coordinates of g'-l are
+ y"ei(cpn+e')s
[-el,
Y',
q'
- 8'
(3.43)
+
7r].
3-6 Unitary Representations of Lie Groups We shall be interested in the connection between the local multiplier representations of T3 and (global) unitary representations of E3 on a Hilbert space. Hence, we make a brief digression to discuss unitary representations. The basic concepts of the theory of unitary representations of (global) Lie groups are presented in several standard references (Naimark [1, 21; Helgason [13). Here, a few of the fundamental definitions rnd results in this theory will be listed which are useful for the study of special functions. It is assumed that the reader is familiar with the basic concepts of Hilbert space theory. Let X be a complex Hilbert space with the inner product of two vectors f, f ' in X denoted by (f,f') and the norm o f f by /-f 1 = ((f, fW2.We aSSume (f, a f ' ) = a ( f , f'), (Uf, f')= d(f, f'> for every a E @, i.e., the inner product is linear in the second argument, conjugate linear in the first argument. A sequence of vectors {f,&}, n = 1, 2,,,.,is said to converge to f EX,(f,,+ f),if limn+mI f, - f I = 0. Let 9 be a vector subspace of X . 9 is dense in X if for every f E X there is a sequence of vectors f, E 9 such that f,, +f. 9 is closed if every convergent sequence of vectors in 9 converges to an element in 9. X is closed by definition. If @ is a subspace of X, 9,the closure of @ is defined to be the intersection of all closed subspaces of X containing a.It is easy to see that 3 is closed. A linear operator U on X is unitary if (Uf, U f') = (f,f')for every f, f ' E X. If U is unitary, so is U-1 where U-l is the unique operator such that UU-' = I, (I is the identity operator). Let G be a real (global) Lie group. A unitary representation of G
3. LIE THEORY AND BESSEL FUNCTIONS
68
on X consists of a family of unitary operators U(g) on X , defined for every g E G, with the properties:
U(gg') = U(g) U(g')
for all g, g' E G, if g,+g in the topology of G then U ( g , ) f 4 U(g)f in X , (2) for all f EX. (1)
Since the unitary operators have unique inverses, it follows easily from ( 1 ) that U(e) = I and U( g-l) = U(g)-l. Property (2) states that the operators U( g) are strongly continuous as functions of g. A unitary representation of G on SEO is irreducible if there is no proper closed subspace of X which is invariant under all the operators U( g), i.e., if there is no proper closed subspace X ' of X such that U(g ) f E X ' for all f EX',gE G. Let '3 be the Lie algebra of G and exp a, a E '3, the exponential map from '3 to G. If the operators U( g) form a unitary representation of G, then in analogy with Section 1-3 we might try to construct a representation of 3 in terms of linear operators L, on SEO defined by
where the limit exists in the sense of the norm of X . However, the limit may not exist for all vectors f so L, may not be well defined as a linear operator on SEO. Denote by 9 ( L , ) the set of vectors f for which this limit exists. Obviously, B(L,) is a subspace of X . Suppose there is a dense subspace 9 of X with the properties:
(1) 9 C 9 ( L , ) for all 3, (2) B is invariant under all the operators U(g), (3.45) (3) 9 is invariant under all the operators L, , (4) For a n y f g 9 the vector U(g)fis an analytic function on G. We say a vector-valued function h of G on X is analytic at goE G if there exists a coordinate system (yl ,...,y m }on a neighborhood W of go such that yl( go) = ... = ym(go) = 0 and h(g) ==
n,.
1
....n,>o
an,**.n,yl(g)n'
yni(gP,
g E W*
Here, the coefficients u , ~ ~ belong . . . ~ ~to X and CIL1,...,n,>O 1 an,...n,I < rn (Helgason [ I ] , p. 440). If such a subspace 9 exists we can immediately carry over the standard methods in Chapter 1 relating Lie groups to Lie algebras. On the dense
3-6. UNITARY REPRESENTATIONS OF LIE GROUPS
69
subspace 9 Theorems 1.8-1.1 1 relating local multiplier representations to algebras of Lie derivatives have exact analogs. Thus, forf E 9 we have
(3.46)
(4)
d dt U(exp tor)f = L,(U(exp tor)f).
-
These results follow from the fact that U ( g ) f i s analytic in g (Naimark [2], Chapter 3). Convergence in (3) is in the sense of the norm. Equations (3.46) show that on 9 the infinitesimal operators Lu completely determine the unitary operators U(g) for g in a neighborhood of the identity. However, since 9 is dense in 2f and U(g) is a bounded operator it follows from a standard argument that U(g) is uniquely determined on S? for all g in the connected component of the identity element in G (Naimark [2], p. 100). I t can be shown that such’a subspace 9 described above actually exists for every unitary representation of a Lie group G (see Helgason [ 11, p. 441, and references given there). Thus it is always possible to use Lie algebraic methods to derive information about (possibly infinite-dimensional) unitary representations of Lie groups. We will not need this result but quote it here to show that the methods used in the rest of this chapter are not as special as they might appear. Lemma 3.1 of G, then
If the operators U(g) form a unitary representation
=
-(f,L,h)
(3.47)
for all f,h E 9 and all or E 9. PROOF Since the operators U(g) are unitary and form a representation of G we have
(U(exp t a ) f , hj
=
(f,U(exp - tor)h)
for all real t. Differentiating both sides of this equation with respect to t and evaluating at t = 0 we obtain the lemma. Lemma 3.1 gives necessary conditions that operators L, must satisfy to be obtained as the infinitesimal operators of a unitary representation of G.
70
3-7
3. LIE THEORY AND BESSEL FUNCTIONS
Induced Representations of g3
T h e above results can now be applied to find unitary irreducible representations of E, . Let U be an irreducible unitary representation of E , on the Hilbert space X and let B be a dense subspace of X satisfying the properties (3.45). First of all, from (3.39) and (3.41) we find
Thus E , is uniquely determined by y1, yz, y,. Equations (3.46) then show that the operators U(g) for all g E E, are uniquely determined by the infinitesimal operators La. Second, if B’ C 9 is invariant under all the infinitesimal operators La of an irreducible unitary representation of E, , then 9’is dense in B. For, if 9’ is invariant under the L a ,it is clear from (3.46) that 3’ (the closure of 9’)is invariant under U( g) for allg E E, . Since the unitary representation is irreducible we have G’ = X. Thus, 9’is dense in both B and X . We can conclude that 9 contains no proper closed subspaces invariant under the La. Define the infinitesimal operators Jk on B by
for all f E 9. These operators satisfy the commutation relations
and determine a representation of 8, on B. Therefore, the operators J* = Jz iJ1 , J3 = iJ3 satisfy the relations
+
Clearly, J*, J3 induce a representation p of the complex Lie algebra F, on 9.We will investigate which of the irreducible representations Q ( w , mo) of F , can be obtained in this way on some dense subspace 9’ of B, i.e., the conditions under which p restricted to 9’is isomorphic to &(w, mob According to Lemma 3.1,
3-8. THE UNITARY REPRESENTATIONS (p) OF
E3
71
for all f,h E 93. Thus,
<JY$h> = - i ( J 3 f , <J+f* h>
=
<(-Jz
h> = + i ( f , J3h) =
+
iJI)f,
h>
Recall that the representation the relations J Y m = mfm
=
+ iJ#>
(3.50) =
(fi
J-h).
(3.51)
m,) of F3was determined by
Q(w,
>
(f, J3h>
J*fm
= wfm*tl
where w # 0, m = m, + k, and k runs over the integers. Now we assume that the vectors f , are in 93 and use (3.50) and (3.51) to find restrictions on o and m,. From (3.50), Thus (6- n) (fn, ,f n ) = 0 for all m,n in the spectrum of J3. If m # n this relation proves (f,, fn) = 0. However, if m = n we have ( f i - m) Ifm l2 = 0. Since f, # 0, the eigenvalue m must be real. Further, from (3.46) we find U(exp Of,)
fm =
exp(-iO
J3) fm
.
= rimym
Since exp(2rr2,) is the identity element of E, , we must have e-*&, = 1 for all m in the spectrum of 5,. Thus, m must be an integer and m, = 0. From (3.51) W
,fm+l>
= (J’fm
rfm+J
1
J-fm+l>
=
4 f m ,fm>
for all integers m. Hence, w/W = I f,+l f, l2 > 0. This relation can be satisfied only if w is real and I f f f l + lI = 1 f , 1. Consequently, all the vectors f , have the same length and without loss of generality we can assume 1 f , I = 1 for all integers m. This analysis proves that any. irreducible representation of F , induced by a unitary representation of E3 must be of the form Q(w, 0) where w is a nonzero real number. Furthermore, the vectors fnz , m an integer, form an orthonormal basis for 93 (hence, for X ) :
3-8 The Unitary Representations (p) of E3 Conversely, we will show by explicit computation that, in fact, all of the representations Q(w, O), w real, induce unitary irreducible representations of E, . (Since Q(w, 0) Q( - w , 0), without loss of generality we
72
3. LIE THEORY AND BESSEL FUNCTIONS
can assume w < 0.) T h e unitary representations of E, can be obtained formally through an examination of the multiplier representation (3.6) of T,. If m, = 0, (3.6) depends on the parameter T only in the form e‘ (if m, # 0 this is not true): (3.6)’
Thus, the operators (3.6)’ define a single-valued multiplier representation of the multiplicative matrix group Ti with elements
(0
cT0
g(T,
b, c ) =
0
eT
0
c
b), 1
b, c, 7 E c.
(3.52)
T h e matrices (3.52) are obtained from the matrices (3.4) of T , by eliminating the first row and last column. Ti is not simply connected. However, as local Lie groups, Ti and T , are isomorphic. Comparing (3.52) and (3.42) we can consider E , as the real subgroup of T i consisting of the matrices (3.52) such that Re T = 0 and b = --E. I n fact,
o G e,
< 2.rr, Y 2 0.
(3.53)
Using this embedding of E, in Ti we can define a multiplier representation of E , directly from (3.6)’ by restricting the group elements to E, :
Since 1 eiez I = I z I for all 8, the representation (3.54) can be restricted to functions defined on the unit circle: z = eia, 0 01 < 2n. T h e basis functionsf,,*(z) = zmrestrict tofm(a) = e i m o for all integers m. Now we have the ingredients required to define irreducible unitary representations of E, . Let S‘ be the complex Hilbert space consisting of all functions f(01), 0 01 < 277, mod 277, such that
<
<
J O
3-8. THE UNITARY REPRESENTATIONS (p) OF E,
73
The inner product on S? is
(3.55) As is well known, the functions m = O , f l , f 2 ,...,
i, ~
fm(a) = e
form an orthonormal basis for S?, =L
(fm ,fn>
We define a representation (p), p on X such that
, n*
> 0, of E,
by unitary operators U(g)
[U(g)f](a) = eipr cos(-)f(,
-
0)
(3.56)
where f E Xand g = g[B, Y,~p]E E, . The expression for U(g) follows immediately from (3.54) by setting z = eiu, p = --w. The operators U(g) are unitary since
for all f, h E X , g E E, . We define the matrix elements of (p) with respect to the basisf, by unm(g) =
=-
9
U(g)fm>
2w
0
exp[+ipr C O S ( ~9') - im0
- jn-mei[(m-n)~-m8]
Jn+,&y),
-m
+ i(m - .)a] < n,
m
da
< a, (3.57)
where the J,reare Bessel functions of integral order. T o justify the last equality note that for z = ieiu,Eq. (3.15) becomes m
e i r COS
O(
=
C
ilJt(y) e i h .
l=--00
Thus the exponential functions under the integral sign in (3.57) can be expanded in a series of Bessel functions and integrated term by term to obtain the indicated result. Note the similarity between (3.14) and (3.57).
3. LIE THEORY AND BESSEL FUNCTIONS
74
By construction, the operators U(g) satisfy the representation property U(g')
= U(g)
W')
(3.58)
for all g, g' E E , . However, we can also verify this directly. I n terms of the coordinates (0, x1 , x2) for g, (3.56) becomes [U(g)f]((~) = exp[ip(xl cos (Y
+ x2 sin .)If(..
-
0).
Thus, if the group elements g and g' have coordinates (0, xl, x2), (O', x i , x@, respectively, we find U(g)[U(g')fl(4 = exP[G4% cos = exp{zp[(x,
+
X;
x2
cos (Y
sin(or
-
= exp{ip[(x; cos
+ sin 4llU(g')fl((Y - 0) + x2 sin a) + (xi cos(a - 0)
0))]}f(a - 0 - 0') 0 - x; sin 0
+ xl)
+ ( x i sin 0 + x; cos 0 + x 2 ) sin =
[U(gg')f](a)
+ 0', X;
cos 0
- 0 - 0')
(Y]}~((Y
for all f EX,
since the coordinates of gg' are (0
cos (Y
- x; sin 0 + x1 , x; sin 8
+ x; cos 0 + xJ.
This proves that ( p ) is indeed a unitary representation of E,. T h e methods introduced in Section 3-7 could also be used to show that ( p ) is irreducible. However, we will give a direct proof of this fact. Lemma 3.2 T h e representation (p), p
> 0, is irreducible.
PROOF We assume (p) is reducible and obtain a contradiction. Thus, suppose there exists a proper closed subspace 9 of X such that U(g)fE 9 for all g E E, , f~ 9.Let P be the self-adjoint projection operator on 9. That is, P is the operator on X uniquely defined by the conditions (a) Pf = f f o r all f~ 9; and (b) Ph = 0 for all h E 91where 9'l
= { h EX: (h,f) =0
for all f~ 9}.
From elementary considerations in functional analysis it follows that (i) (Ph', h") = (A', Ph") for all h', h" EX,i.e., P is self-adjoint; (ii) P2 = P; and (iii) U(g)P = PU(g) for all g E E3 (see Naimark [I], Chapters 4,6). Furthermore P # 0, I, since 9 is a proper subspace of X . Let gobe the element of E, with coordinates (0, 0,O).Then U(g,)f,,, = cirnsf,wheref,, is a basis vector for X . From (ii) we obtain U(ge)Pfm=
3-9. THE MATRIX ELEMENTS OF (p)
75
for all integers m. Since the vectors f, form a basis for ,W we must have Pf, = 01, f, , a, a constant. T h e property P2 = P implies C Y = ~ a m , hence am = 0 or or, = 1 for each integer m. By hypothesis there exist nonnegative integers m',m" such that Pfm*= 0,Pfmn = f,n. Thus, Umfrn.(g) = (fm' U(g)PfmH) = ( f m ' PU(g)fmn) = (Pfm' U(g)fmn) = 0 for all g E E, . However, from (3.57) we see that no matrix element Un,(g) of ( p ) is identically zero. This contradiction proves the lemma.
e--im@Pf,,
9
3-9
3
9
The Matrix Elements of (p)
We can obtain information about Bessel functions of integral order from expression (3.57)for the matrix elements of (p). Since the operators U(g) are unitary and form a representation of E, , the matrix elements satisfy Unm(g-')
=
umn(g),
or
JnW.
J-n(r) =
(3.59)
Moreover, U,,(e) = an,, where e is the identity element of E,. In terms of Bessel functions this implies if n # 0,
Jn(0)= 0
],(O)
=
1.
(3.60)
Using the Schwarz inequality, we have
I unm(g)I
=
I(fn
>
< Ifn
Uk)fm>l
I . I U(g)fm I
= Ifn
I * Ifm I = 1
<
I Jn(d 1. Since U(g'g")= U(g') U(g") for all g', g"E E, the matrix elements satisfy the addition theorem so
unm(g'g")
m
=
unk(g')
ukm(k!n).
(3.61)
k---a,
Substituting expressions (3.57) for the matrix elements and simplifying we obtain m
e-im'Jm(y)
=
C
e - W V + ( m - k ) ~ ' l Jm-k(y') Jk(y")
(3.62)
kc--ao
where ieiq = r'eiw' case of (3.32).
+ r"eiq". This
is Graf's addition theorem, a special
76
3. LIE THEORY AND BESSEL FUNCTIONS
If
v'
I' = Y" = x,
=
0, and q" = T,then r = 0, v = 0, and m
C
k=-m
JIc+n(X)
.Id.)
= 8n.o
where we have used (3.59) and (3.60). = 0, r = r' + Y", and (3.62) becomes = T'' = 0 then If Jm(7'
+
m
I") =
C
k=--m
Jm-dy')
J k ( O
Finally, we can derive a useful integral formula for the product of two Bessel functions from (3.62) by setting v' = 0 and multiplying by einc". Integrating both sides of the resulting equation with respect to cp" we obtain
where
reiq =
rf
+ r"ei""
3-10 The Infinitesimal Operators of ( p ) T h e infinitesimal operators of the representation ( p ) defined by (3.49) are easily computed to be Jl = i p cos a,
. .
Jz = zp sin a,
= -
J3
a z.
(3.64)
These operators are all defined on any dense subspace 9 of 3? satisfying properties (3.46) and they leave B invariant. Forming the operators Jf, J-, 53 in the usual manner, we have J+
w&,
J-=
we-i~,
J3
=
a -i -, aff
w = -p.
T h e action of these operators on the analytic basis vectorsf,,(a) = eima is given by J+fm
= wfm+l,
J-fm
=
wfm-1,
J"fm
= mfm
in agreement with Eqs. (3.1) for the representation Q(w, 0) of F3 . Thus, each representation Q ( w , 0), w < 0, induces an irreducible unitary representation ( - w ) of E3 .
3-10. THE INFINITESIMAL OPERATORS OF (p)
77
As in Section 3-2 we could derive recursion relations and differential equations for the matrix elements of (p). However, we shall omit this as the results are merely special cases of the corresponding results in Section 3-2. Similarly the addition theorem (3.62) is a special case of the addition theorem (3.16). From the general theory of Lie groups it can be shown that up to unitary equivalence all of the irreducible unitary representations of E, are of the form (p), p > 0, except for the trivial 1-dimensional representations x,, , n an integer, defined by xn[g(O, XI 9
XPll = elne
(see Bingen [l], Vilenkin [l, 31). Thus, the Lie algebraic methods presented in this chapter suffice to find all the irreducible unitary representations of E, . T h e reason for this is essentially topological: T h e l-parameter subgroup C = {exp Of,} of E, is compact. Because C is compact, the Peter-Weyl theorem guarantees that any unitary representation U of E, when restricted to C can be decomposed into a direct sum of 1-dimensional irreducible representations of C. Since the 1-dimensional representations of C are of the form p,,(exp O$,) = ein@,n an integer, it is always possible to find a basis for the representation space of U consisting of eigenvectors of the operator J, = -iJ3. If U is irreducible it is not difficult to show that each eigenvalue has multiplicity one. Thus, the hypotheses (2.19) of Section 2-6 are satisfied and our methods succeed in determining all unitary irreducible representations of E , . Another real Lie group whose Lie algebra is a real form of F, is T,". The elements g of T,"are those matrices of T, which are real:
In contrast to E, , the group Tf has no compact 1-parameter subgroups. In this case the Peter-Weyl theorem will not help and, as the reader can verify for himself from Lemma 2.1, none of the representations &(w, m,) of 4 induce unitary representations of Tf.T h e importance of T," in special function theory is related to the study of integral transforms and is beyond the scope of this book (see Vilenkin [3]).
CHAPTER
4 Lie Theory und Confluent Hypergeometric Functions
I n the last chapter fundamental properties of the cylindrical functions were deduced by relating those functions to the representation theory of 9(0,0). Here, analogous arguments will be used to study confluent hypergeometric functions as they relate to the representation theory of 9 ( 0 , 1). Besides treating the general confluent hypergeometric functions lFl(a; b; x), we shall also consider certain special cases: the Laguerre polynomials L,"(x), the parabolic cylinder functions DnE(x),and the Hermite polynomials H,(x). These special cases arise naturally. I n Section 4-11 we will introduce a real (global) Lie group S, whose Lie algebra is a real form of g(0, 1). We will examine the relationship between the irreducible representations of g(0, 1) and the unitary irreducible representations of S, . This study leads to addition theorems and orthogonality relations for the Laguerre polynomials. T h e problem of decomposing tensor products of unitary representations of S, into irreducible parts will also be considered. T h e solution of this problem leads to identities expressing the product of two Laguerre polynomials as a sum of Laguerre polynomials (Section 4-17), and as an integral over Bessel functions (Section 4-19). Furthermore, the product of a Laguerre polynomial and a Bessel function can be expressed as a sum of Laguerre polynomials (Section 4-20). Each of these expansions will be given an explicit group-theoretical interpretation. Finally in Section 4-21 we will use the fact that g(0,O) is a contraction of g(0, 1) to derive a formula for Bessel functions as limits of Laguerre polynomials.
4-1. THE REPRESENTATION R(w. m, , p)
79
Note: There is another real form of 9 ( 0 , 1) (the Lie algebra of all real matrices (1.30)), distinct from the one we consider here. A study of the representation theory of this new real form leads to integral identities involving confluent hypergeometric functions (see Vilenkin [3]). 4-1
The Representation R(w, m, , p)
The irreducible representation R(w, m, , p ) of 9 ( 0 , 1) is determined by complex constants w , m,, p such that p # 0, 0 Re m, < 1, and w + rn, is not an integer (Theorem 2.2). T h e spectrum of this representation is the set S = {m, n: n an integer},
<
+
and the representation space V has a basis {f,}, m E S, so that JYm J-fm
= (m
= mfm
+
Efm
9
w)fm-19
= pfm
9
= (J'J-
c,,,fm
J'fm
= tLfm+l*
- EJs)fm
= pwfm
-
(4.1)
The commutation relations satisfied by the infinitesimal operators are
[I3, 11'
=
&J*,
[I+,J-I
[I*, E]
--E,
=
=
[I3,E] = 0.
(4.2)
We will find a realization of this representation in such a way that the operators J*, J3, E become the linear differential operators (2.36) acting on a vector space of functions of one complex variable, z. Let the representation space Vl be the complex vector space consisting of all finite linear combinations of the functions h,(z) = zn,n = 0, f l , &2, ..., and set A = m, , cg = m, + w in Eqs. (2.36) to obtain the operators
on V l . Define the basis vectors m = m, n for all m E S. Then
+
JYm= (m,+ z
"1
of Vl by fm(z) = h,(z) where
fm
+ n) zn = mfm ,
zn = (m,
J+fm
= (pz) Z" = p ~ " + l= tLfm+l*
J-fm
=
Efm
( & + -)dzd Z
= tLfm
Zn =
(m
+ w ) zn-1
7
and we have a realization of R(w,m, , p).
=
(m
+w)
fm-l,
80
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
As in Chapter 3 we use this realization by differential operators to obtain a local multiplier representation of the local group G(0, 1) whose Lie algebra is 9 ( 0 , 1). The matrix elements of this local representation with respect to the basis { fm} will be confluent hypergeometric functions. Although the realization of R(w, m, , p) given above was constructed from the differential operators (2.36), it could also have been constructed from the operators (2.37). As far as the computation of matrix elements is concerned it makes no difference which operators we use. T h e matrix elements are uniquely determined by the Lie algebra relations (4.1) and do not depend on the particular realization of these relations. In Section 1-2 it was shown that 9(0,1) is the Lie algebra of the local Lie group G(0, 1):
We can extend the realization of R(w, m, , p ) defined on Vlto a local multiplier representation of G(0, 1) defined on a,, where QIl is the complex vector space of all functions of z analytic in some neighborhood of the point z = 1. Clearly 0, 3 Vl and QIl is invariant under the operators
According to Theorem 1.10, these operators generate a Lie algebra, isomorphic to 9(0,I), which is the algebra of generalized Lie derivatives of a multiplier representation A of G(0, 1) acting on QI, . We will compute the multiplier v explicitly. From the theorem, the action of the I-parameter subgroup {exp c f - , c E fZ'} of G(0, 1) on a, is obtained by solving the equations
_ dz - 1, dc
d
dc v(z0,
exp c f - )
m +w =0 v(z0, exp c$-) 2
with initial conditions z(0) = 9 # 0, v ( 9 , e) = 1. Here, e is the identity element of G(0, 1) and exp c f - is given by (1.33). The solution of the differential equations is
4-1. THE REPRESENTATION R(w, m, , p)
Thus, iff
E
81
GI, is analytic in a neighborhood of 20 then
and A(exp c f - ) f is an element of domain off. Similarly we obtain "exp
a, for
1 cjz" I < 1,
20
+ c in the
bf+)fI(z") = exp(PWf(z"),
' ) emoTf(eTz"), [A(exp ~ f ~ ) j ] ( z =
[A(exp a & ) f ] ( z o )= ewf(z").
If g E G(0, 1) has coordinates (a,b, c, g
= (exp
T)
it is easy to show that
bf+)(exp c$-)(exp
.f3)(exp a&).
Thus, for I c I, I T I sufficiently small, the operator A(g) acting on f E QI, is given by [A(g)fl(z) = M e x P by+) W X P c f - ) AlexP 7 f 3 )N e x P aS)fI(z)
T o make sense out of this expression we restrict g to the open set M C G(0, 1) where M = {g E G(0, 1): I c I < I}. As a local Lie group, M is isomorphic to G(0, 1). For everyfE 6Zllet D, be the domain off, i.e., the open set in containing 1, on which f is defined and analytic. Define W,(g), the domain ofg E M, by &?,(g) = { f Q~ I, : eT(1 c) E D,}. It follows from these definitions that for any g E M and f E W,(g) we have A(g) f E QIl, where A(g)f is given by (4.5). I n fact, DA(g)f = {z E G ': I c / z I < 1 and eT(z + c) E D,}, so 1 E DA(0)f. (The factor (1 ~ / z ) ~ o + in w (4.5) is defined by its Laurent expansion about z = 0.) The problem of properly defining the domain of the operator A(g) on QIl is analogous to the problem of defining the domain of an unbounded operator on a Hilbert space. We can given a precise interpretation of the representation property of the operators A(g) as follows: If g, ,g2 , g,g2 E M and f E QI, such that f E W,(gz) and A(g2) f E al(gl),then f E ~ d g 1 g z and )
c,
+
+
A(g1g2)f = A(g,)"g,)fl
Ea 1
-
(4.6)
The problem of properly interpreting (4.6) is analogous to the problem of defining the product of two unbounded linear operators on a Hilbert space.
82
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
The elements of V, are analytic for all z # 0 so for any f E V, , g E M , the expression A ( g ) f is well defined as an element of 02, . I n fact Da(p)f= { z E f2: I z I > I c I} 3 {I}. Consequently, if h E 7, then Dh = {I z I > dh} where dh is a constant such that 1 > dh 20. (Recall from Section 2-2 that 9, is the space of all finite linear combinations of functions of the form A ( g )f where g E M and f E 6 . By definition, 7, is invariant under A.) Thus, if h E 7, then h has a unique Laurent expansion m
h(z) =
1 anzn,
an€
n=--co
e:
which converges absolutely for all I z I > dh . According to the above results the subspace 9, is invariant under A, and the basis functions fnt(z)= h,(z) = zn,m = m, n, for *y; form an analytic basis for 7 , (see Section 2-2). As usual the matrix elements A,,(g) of the operators A ( g ) on TI are defined by
+
m
[A(g)hJ(z)
=
C
l=-m
At&) h i ( z ) ,
g E M , k = 0, &1,--.,
(4-7)
or
I c/z 1 < 1.
(4.8)
(4.9)
we obtain the addition theorem Atkk1g2)
m
C
=
Atj(g1) Ajk(g2),
I,
= 0,
*2,.** *
(4.10)
j=-m
Equation (4.9) is defined only for those group elements g , ,g , E M such that g,g, E M and A(g,)h, E W,(gl).However, we shall soon see that the matrix elements A,,(g) are entire functions of the complex group parameters (a,b, c, T) of g ; hence, the matrix elements can be defined by analytic continuation for all g E G(0, 1). Moreover, the group parameters of g,g, are entire functions of the parameters of g , and g , : &?,(a1 > 61 > C l
9
71)
= glg2(al
g2(a, > b,
9
c2
9
72)
+ a, + clb2eT5b, + eT1b2,c1 + e-'lc,
,T~ + T ~ ) . (4.11)
83
4-1. THE REPRESENTATION R(w, m o tp)
Thus, by analytic continuation the addition theorem (4.10) is valid for all g1 ,g, E G(0, 1). To derive an explicit expression for the matrix elements A,,(g) we expand the left-hand side of (4.8) in a Laurent series in z and compute the coefficient of z,. T h e result is
where the sum extends over all nonnegative integral values of s such that the summand is defined. For convenience we have set p = m, + W . Comparing with (A. 10) we have
*
Az,(g)
=
lFl(-p
exp[p
. ,F,(-p
-
I; K
-I
+ 1 ; -&)
if k 2 I ,
-K
+ 1; - 4 c )
if
+ (m,+ k M ( t ~ b ) ~ - ~ (1 --PY
-
K; I
12
(4.12)
K.
Here, p is not an integer. The functions are the confluent hypergeometric functions defined by their power series expansions. I t is clear from (4.12) that the matrix elements are entire analytic functions of the group parameters. Perhaps the most convenient form in which to express the matrix elements is in terms of the generalized Laguerre functions L r ) , (A. 15): A Zk (g)
= e ~ 5 + ( m o + k ) ~ ~ k - p+z Z L ( k (-pbc), -1)
k, 1 integers.
(4.13)
Substituting (4.1 3) in (4.8) and simplifying, we obtain the generating function e-clt(l
+t
m
)= ~
1 tlLjyL(c),
O < 1 t I < 1.
(4.14)
Z=-cc
#
Note that the Laguerre polynomials L t ) , n an integer, do not occur as matrix elements of R(w, m, , p). Laguerre polynomials will arise in the computation of matrix elements of TU,@ and la,,+to be considered later. We can obtain addition theorems for the Laguerre functions by substituting (4.13) into (4.10). After some simplification there results e-clbz(cl
+
c
+ b,)(c, + 4 1
m
=
j=--03
(Cl)j+n
L;j+"'[b,C,](c,)-jL;;;;,[b2C2]
(4.15)
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
84
where n is an integer, p is not an integer, and b, , b, following limits can be verified from (4.14): (0
if n if n
(-n)!
where n is an integer. Therefore, if (4.15), we obtain
, cl, c, E f2. T h e
> 0,
< 0, (4.16)
c1 =
b, = 0, c2 = 1, b, = x in
which is the power series expansion for the generalized Laguerre functions. If c1 = 0, c2 = 1, b, = x, b, = y in (4.15), one obtains
while for c1 = 1, c,
If c1
= -c,
=
=
1, b,
0, b, = x, b,
= x,
=y,
b, = -y we have
c (-1)jL;-")(x)L'-? W
eY (Y - x)" n!
=
p+3-n
(y),
n 3 0,
j=-m
and m
0=
1 (-l)jLbj+"'(x)L~;::,(y),
n
> 0.
j=-rn
As a final example, set c1 = c,
=
1, b,
= x,
b, = y to obtain
rn
e-VL;)[2(x
+ y ) ] = C L ~ + " ) (L:&( x) j=-
y).
4-2. THE REPRESENTATION
4 2 The Representation p
TU,@
85
tamp
The irreducible representation of 9(0,1) is defined for each @such that p # 0. The spectrum S of tw,@ is
W,
E
S
= {-w
and there is a basis { fm , m properties J Y m = mfm
=
J-fm
(m
+
3
+ n: n a nonnegative integer} E
S> for the representation space V with the Efm
wIfm-1,
= tLfm
G.1fm
J'fm
9
=
(J+J-
== tLfm+l>
- E J S ) f m = tLwfm
*
(Heref-,, = 0, so J-f-w = 0.) As in the last section, we can construct a realization of this representation such that J*, J3, E take the form of linear differential operators (2.36) acting on a space of functions of m e complex variable. Namely, we designate by Yzthe space of all finite linear combinations of the functions h,(z) = zn, n = 0, 1, 2, ..., and define operators on V2by setting X = -0, c3 = 0 in (2.36): J3
=
-W
+z-& d '
E
= I.19
J+ = pz,
J-
=
d dz .
(4.17)
The basis vectors fm of V2are defined byf,(z) = h,(z) = zn where m = --o n and n 2 0. These operators and basis vectors satisfy the relations
+
J Y = ~ (-w
zn
= (-w
+ n ) = ~ mfm, (4.18)
for all m E S. Relations (4.18) yield a realization of Tw,p. We can extend this realization to a multiplier representation of G(0, 1) defined on a,, the complex vector space of all entire analytic functions of z. Obviously, GZ2 is invariant under the operators (4.17) and contains Y2as a subspace. According to Theorem 1.10, the operators (4.17) induce a local multiplier representation B of G(0, 1) acting on the space of all functions analytic in a neighborhood of z = 0. However, it will turn out that B(g)fE OZz for all f~ GZz , g E G(0, l), so without loss of generality, we can restrict B to GZz .
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
86
Since the differential operators
+
are formally identical with the operators (4.2) for the case m, w = 0, the action of G(0, 1) on GZ, induced by these differential operators is obtained from the corresponding result (4.5) in Section 4-1 by setting m, = - w . Thus, we have operators B ( g ) defined for all g E G(0, I), such that [B(g)f](z) = e p ( b z + a ) - w f(eTz + eTc), (4.19) f E a, .
+
T h e multiplier v is given by v ( z , g ) = exp[pbz pa - w ~ ] Clearly, . if f E GY2 , i.e., iff is an entire function of z, then B(g)f is an entire function of z ; therefore the space is invariant under the operators B(g). Furthermore, the operators B(g) are defined on GZ, for all g E G(0, 1) and satisfy the representation property
a,
(4.20)
B(g1gz)f = B(g,)[B(gz)fl
for all f E GZ2 and all g, ,g , E G(0, 1). Equation (4.20) is a consequence of Theorem 1.10. However, it is simple to verify its validity directly. Thus, if g, and g, have coordinates ( a , , b, , c1 , T , ) and (a,, b, , c, , r,), respectively, then g,g, has coordinates ( a , a, eT1clb2, b, e%, , cl e-'lc,, r1 r,) and we find
+
B(g,"(g,)fl(z)
+
+ +
+
=
exp[CLb,z
= exp[p(b,
+ Pal
-
+
WT11[B(g2)fl(eT1z
e71c1)
+ e71b,)z + ~ ( a +, a2 + e'1clh) + (c +
- f(e(Ti+To)z
e(Ti+Tz)
1
4 7 1
+ .,>I
e-Tlc2)>
= [B(g1gz)fl(z) for all f E a,. Every function f in has a unique power series expansion
a,
m
f(z)
=
1 anz",
?l=O
an E
C,
convergent for all z E @. Thus, the basis functions h,(z) = zn,n 3 0, of V, form an analytic basis for a,. With respect to this analytic basis the matrix elements B,,(g) are defined by m
[B(g)h,l(z) =
C Bdg) ht(z),
1=0
g E G(0, 11, k = 0,1,2,***, (4.21)
la,+
87
1 B,,(g)zl. =o
(4.22)
4-2. THE REPRESENTATION
or, from (4.19), ep(bZ+O)+(k-wh
(z
+ 4,
W
=
1
T h e group property (4.20) leads to the addition theorem m
Bt,(g,gz)
=
C BtjW Bjkz),
Z 0, g19 g2 E G(O, 1)- (4.23)
1,
G O
T h e matrix elements B rk(g)are easily determined by expanding the left-hand side of (4.22) in a power series in z and computing the coefficient of 21. T h e result is
k, 1 2 0, (4.24) where the (finite) sum is taken over all nonnegative integral values of s such that the summand is defined. We can express these matrix elements in terms of known special functions. Comparison of (4.24) with (A. 10) yields
if l > k > , O .
T h e matrix elements can also be written in the convenient form B, , ( g) = e w z + ( k - w i . r C k - 2 ~ (1k - 1 )
(
-
I
4
9
k, 12 0,
(4.26)
where the functions Lp' are the associated Laguerre polynomials. Substituting (4.26) into (4.22) and simplifying, we obtain the generating function (4.27)
Note the great similarity between the expressions for matrix elements of t,, and the matrix elements of R ( w , m, , p ) given by Eqs. (4.12), (4.13). T h e matrix elements computed here can be obtained from the earlier results by setting p = 0 and restricting the indices I, k to nonnegative integral values.
88
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
I n accordance with our usual procedure, we can obtain identities for the associated Laguerre polynomials by substituting (4.26) into the addition theorem (4.23). T h e results are (after simplification)
(4.28) j-0
+
where n, I are integers, I >, 0, I n 2 0, and b, , b, , c1 , c2.€ C?. Comparing (4.26) and (4.24) we can easily derive the relations if n
CnL~)(bc)I,=,
=
I(
if n
> 0,
< 0,
(4.29) n + l c n
n
if n 2 0 , if n
0
< 0.
These relations can be used to derive identities for the associated Laguerre polynomials which are simple consequences of (4.28). Thus, if c1 = b, = 0, c2 = 1, b, = x, (4.28) reduces to
which is the series expansion for the associated Laguerre polynomials. For c, = 0, c2 = 1, b, = x, b, = y , Eq. (4.28) reduces to
and if c1
=
1, c, = 0,b, = x, b,
= y,
it becomes
4-3. T H E REPRESENTATION J,*+
For c1 -- -c,
=
1, b,
= x,
b,
89
the resulting expressions are
= --y
for all 1 2 n 2 0, and
for all
I, n 2 0. If c1
= c2 = 1, b, = x,
e-112nL'n)[2(x
b,
= y,
we obtain
+y)] = 1L y y x )L;+n-j'(y). m
j=O
4-3
The Representation
Jw,r
As was stated in Theorem 2.2, the irreducible representation lo,, of w , p E @ such that p # 0. T h e spectrum of I,,,,is the set S = {-w - 1 - n: n a nonnegative integer},
9(0,1) is defined for each
and there is a basis { fm , m J"fm
= mfm J-fm
9
Efm
= t.frn-19
E
S } for the representation space V such that J'fm
== -t.fm, Co,,fm
=
=
-(m
(J'J- - EJ3)fm
+ + l)fm+l, w
(4.30)
= -wfm-
(We setf-, = 0, so J+f--, = 0.) I t is impossible to find a realization of this representation in terms of differential operators of the form (2.36). I n fact, if J+ = pz it is impossible to find a nonzero analytic functionf-,,(z) such that J+f-,-, = 0. We could find a realization in terms of the operators (2.37). However, it is more convenient to use the following modification of (2.36):
As is easily seen, these operators satisfy the commutation relations (4.2), hence they generate a Lie algebra isomorphic to 9(0,1). We will define a realization of I,,, such that J*, J3, E take the form of the differential operators (4.31) acting on the vector space V,of all finite linear combinations of the functions h,(z) = z", n = 0, 1, 2, ... . Namely, we define the
90
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
basisvectorsf,of T h e n we have
Vzbyfm(z) = h,(z)
=
znwherem = --n
-
w -
1 E S.
for all m E S. Comparison with (4.30)shows that our choice of differential operators and basis vectors leads to a realization of J u . p . As in Section 4-2 we can extend this realization to a multiplier representation of G(0, 1) defined on the complex vector space B2of all entire analytic functions of z. Since the details of the computation are so similar to the computation for Iw+ it will be sufficient to merely list the results. T h e generalized Lie derivatives (4.31) induce a multiplier representation of G(0, 1) defined by operators C(g),g E G(0, l), on L I Z :
for all f E Bz, z E @. T h e representation property is
and the matrix elements C,,(g) are defined by
or
g E G(0, l), K 2 0.
T h e addition theorem for the matrix elements is
(4.34)
4-4. DIFFERENTIAL EQUATIONS FOR MATRIX ELEMENTS
91
From the generating function (4.34)we can obtain the following explicit formula for the matrix elements: C,,(g) = exp[p(bc
- Q) - (w
+k +
k, 1 3 0. (4.36)
l)T]bk-z Lik--”(-&),
We can derive addition theorems for the associated Laguerre polynomials by substituting (4.36)into (4.35).However, the resulting expressions are identical with expressions (4.28)derived in the last section, so we will omit this computation.
44
Differential Equations for the Matrix Elements
T h e individual matrix elements of the representations R(w, m, , p), , and J, are all entire functions of the complex group parameters a, b, c, T and can be considered as analytic functions on the group manifold of G(0, 1). Let GI[G(O, l)] or Ol for short, be the complex vector space of all entire analytic functions on G(0, 1). There is a natural action of G(0, 1) on GI as a transformation group. For every g’ E G(0, 1) we can define a linear operator P(g’):a + a by
t,,,
(4.37)
From this definition it is evident that P(glg2)f = P(gl)[P(g,)f] for all gi 9 g2 E G(0, 1). Clearly each of the matrix elements Ajk(g),Bjk(g),Cjk(g)computed above is a member of GI. Furthermore, the action of P on these matrix elements is easily determined. We have
c A,&’) m
[P(g’)A,,l(g)
= A,,(gg’) =
I=-m
A,Z(d,
j , k integers, m
[P(g‘)B,,I(g)= Bjd&)
1 B&’)
= I=O
j,
Bdg),
k nonnegative integers,
m
[P(g’)Cj,I(g)
=
Cjk(gg’) =
C Cldg’) Cjdg), z=o j , K nonnegative integers.
(4.38)
92
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
Comparing these expressions with (4.7), (4.21), and (4.33) we see that for fixed j , the functions {Aik}form a basis for the representation R(o, m, , p), the functions {Bjk} form a basis for the representation t o , r , and the functions { c j k } form a basis for the representation J w , p . Therefore, the Lie derivatives J+, J-, J3, E defined on @ by
must satisfy the commutation relations (4.2), and their action on the matrix elements Ajk, Bjk, c j k must be given by
J3cjk(g) = ( - W J-Cjdg)
- 1 - k) Cjdg), = tLCi
k+l(g),
Co.1Cj&)
=
J+Cjkk) = k c j k--l(g)r
EC&) -PWCj&)
= -PCjdg),
(4.42)
for all nonnegative integers j, k.
These expressions yield recursion relations and differential equations for the matrix elements. To evaluate them we must compute the Lie derivatives J*, J3, E defined by (4.39). In terms of the usual coordinates (Q, b, c, T ) for elements of G(0, 1) the Lie derivatives will be linear differ-
4-4. DIFFERENTIAL EQUATIONS FOR MATRIX ELEMENTS
93
ential operators in these four complex variables. However, the confluent hypergeometric functions and Laguerre functions associated with the matrix elements Ai,, B j , , Ci,are functionally dependent only on the product bc of the group parameters. For this reason it is convenient to adopt new coordinates [a, I, c, r] where a, c, r are defined as before, but Y = bc. This new coordinate system is not uniquely defined over the whole group, but only for those group elements such that c # 0. If g E G(0, 1) has coordinates [a, Y , c, T] and
0
0
0
then the coordinates of gg‘ are [a
+ a’ + evcb‘, r (I + e-Tc’ +
b‘c’
1
+ eTb‘c ), c + e-qc’, + r 7
7’
3
in the new coordinate system. Using this result and the-definition (4.39) of the Lie derivatives, we obtain J+ = eTc
(a, a +ay ,
J-
=
e-7
a
a
r a (-aca + --j, c ar (4.43)
E = -aa ’
J 3 =z 7
C0,I
=r
a2
p
a +ar + c -ac ar ‘ m+c--- acaa a2
a2
a2
a2
ar aa
+
*
In terms of the new coordinates, A j , and B j , are given by A,,(g)
= e’a+(m,+P)sCX.-iL(~-;i) p+j
p = m,
(-PI),
+
w,
B,,(g) = e’a+(-w+k)rc~-jL(~-j) j (-P+
Substituting these expressions into (4.40), (4.41) and simplifying we obtain the relations
-aq y Y ) ar
- L ( y y )= -LW+l)(y), P
P
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
94
valid for p E @, k an integer. I n fact, these relations follow from (4.40) when p is not an integer and from (4.41) when p is an integer. Similar results can be derived for the matrix elements Cik(g).However, we will omit this computation.
4 5 The Representation to,,+,6I,,,,, As was demonstrated in Section 4-2, the representation Tw,, , p # 0, of 9 ( 0 , 1) can be realized on the space V, of all finite linear combinations of the basis functions hk(z) = zk, k = 0, 1, 2, ... . I n terms of this basis the matrix elements Bj;*")(g),g E G(0, I), are defined by (4.26). ( T h e superscripts have been added to denote the particular representation To,+ .) We will define a representation to,,,, Q To,,+, of 9(0,1) on the space V, @ V, of all finite linear combinations of the basis functions h,,,(z, w ) = z k w l ,k, 1 = 0, 1, 2,... . Note that Vz@ Vzis contained in the vector space GZ, Q GZ, consisting of all entire functions f ( z , w ) in the complex variables z and w . T h e functions hk,,(z,w ) form an analytic basis for GZz Q 02, since every element f of this space has a unique power series expansion
c m
f(Z,w ) =
UL1ZLW1,
E
@,
4
7
Ukl
k.l=O
convergent for all z, w E IT. T h e operators T(g), [T(g)fl(z, w )
= exp[b(plZ
+ pzw) - (ul +
+ eTc, e7w + eTc),
= g(u,
6 , c,
T)
E
( ~ 1
f~ aZ@ 02, ,
*f(e7z
where g
+ + PZ)QI (4.45)
G(0, l), clearly satisfy the group property
T(g,gz)f
= T(g1)"m)fl
for all g, ,g, E G(0, 1) and thus define a multiplier representation of G(0, I ) on Q GZ, . This multiplier representation of G(0, 1) induces a representation of "(0, 1) in terms of the generalized Lie derivatives
a,
J3 = - ~ 1- W Z
a +wa , +zaz aw
+
J' = p l ~
~ Z W ,
(4.46)
T h e representation of 9(0,1) on VzQ Vzdefined by these operators, which we denote t,,,,, @ f o e , r 2 , is not irreducible. We will decompose
4-5. THE REPRESENTATION
T,,, 0T,.,
95
V2@ V2into a direct sum of subspaces such that each subspace is irreducible under the action of Y(0, 1) as given by (4.46). Successful completion of this task will enable us to derive formulas expressing the product of two associated Laguerre polynomials as a sum of Laguerre polynomials. I n order to carry out this decomposition, however, we must require p, , p2 # 0, pl p2 # 0. Let Q[.,.] be the complex symmetric bilinear form on V2@ V2 defined by
+
for all a1 , u2 E @, f, ,f 2 ,f3 E V2@ V2. Equation (4.47) together with (4.48) completely determines Q. We will use this bilinear form as a _. bookkeeping device. Lemma 4.1 If J+, J-, f 2 E V2@ % , then
J3,
E are defined by (4.47) and f, ,
(i) Q[J”f, 9 f i l = Q[fi JYJ, Q[fi J-f21, (ii) Q[J+fl9fil (iii) Q P f I , f 2 3 = Q[fi Ef21. PROOF Properties (i) and (iii) are trivial. Since Q is bilinear, it is and sufficient for the proof of (ii) to consider the case wheref, -f2 = h,,,,, . From (4.47) we have 9
9
1
9
Q[J+h,.z , hk’.2’]
+
= Q[(P~z ~ 2 wzkw’, ) z“~”] = plQ[z”+’w’, z“w“]
which proves (ii).
+ p 2 Q [ ~ k ~z“w”] 2+1,
96
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
T h e bilinear form Q was defined by (4.47) just so Lemma 4.1 would be valid. We will decompose V2@ Y2into subspaces, each of which forms the basis space for a representation tw,rof 9(0,1). One way to determine this decomposition is to compute the eigenvectors f of J8 which are of lowest weight, i.e., the eigenvectors of J3 such that J-f = 0 where J8 and J- are given by (4.46). Thus, we look for solutions of the equations
a a +zaZ + w -)f(z, aw
J3f(z,W ) = (-q - w2
W)
= hf(z,w ) ,
where the eigenvalue h is to be determined. To within an arbitrary constant the solutions are the functions I n fact,
fs,o(z, w) J”fs.0
= (S
= (z - W ) * ,
- ~1
-4 f s . o
,....
(4.49)
= 0.
(4.50)
s = 0,1, 2 9
J-fs.0
From (4.47) and (4.49) it is a straightforward computation to obtain the result (4.51)
+
(Recall the assumption p1 p2 # 0.) We now define basis vectors f8.k E Y2@ V2 Iby fs.k =
(PI
(4.52)
k)-*(J’>kfs.~
for all integers s, k >, 0. It is easy to see that each function fs,k is a polynomial of order s k in z and w. Moreover, for every positive integer I there are exactly I 1 functions fs,k such that s k = I, i.e., there are I + 1 such functions which are of order I in z and w. Hence, if the set { fs,k} is linearly independent it forms a basis for V2@ v2and an analytic basis for @ .
+
+
a, a,
+
4-5. THE REPRESENTATION
twl+,0 tw2+
+
97
+
-
PROOF (i) J'-f8.k = (PI P2)-k((J')k+fs,o = (PI p2)fS.k+l 0 is easily proved (iii) The identity [J3, (J+)k] = k( J+)k for all K by induction. Then J"fd.k
>
+ = (pi + P~)-'(J+)~ + (pi + = + s + K)(A + = +s + -
= (PI
J3(J')"fs.o
~2)-'
J"fs.0
(-wi
"2
(-w1-
~2
1 1 2 ) ~4J')'fs.o ~
~2)-~(J+)~fs.o
h)fa,k
(iv) Obvious. (ii) We use induction on K. From (4.50), J-f,,, = 0, so the equation is vaiid for k = 0. Assume it is valid for K k, . Then J-fs,k,+i (pi ~2)-'J-J+fs,k,from property (i). But, J-J+ = J+JE, so from (I), (iv) and the induction hypothesis we have
7
<
+
+ = + (A+ = (ko + l)(/%+
J-J+fs.r, = J+J-fs.k,
koJ+fs.k,-l
Efs.k,
~2)fs.k~
/+)fs,k.
+
+
-.
Therefore, J-f8,k,+l = (KO l)fs.k,. Q.E.D. Lemma 4.2 implies that for a fixed value of s, the vectorsf,,, , K = 0, 1, 2, ..., form a basis for the representation fw,+02-8,rl+rl of 9(0,1) (compare Lemma 4.2 with Eqs. (4.18)). Thus, the action of G(0, 1) on the vectors f 8 . k is m
[T(g)fSJ(z, w )
=
z=o
B:p+Ws-s+l+*r) (g)fa.dz,w)*
A, s 2 0, g E G(O, I),
(4.53)
where the operator T(g) is defined by (4.45) and the matrix elements B\g+"*-8*~1+p~) (g)are given by (4.26) (w = w1 wp - s, p = pl p2).
+
+
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
98
Since
by (4.51), it is an easy induction argument to obtain Q [ f s , k , f s , k ] = Msk for all s, k 3 0. T h e property Q[J”f,,k ,f s ’ , k ’ ] = Q [ f y , ,kJ”fs,,k,] implies (S k - Sf - k’) Q [ f s , k , f s , , k , ] = 0. Hence, Q [ f s , k , f s * , ~ = * ] 0 Unless s k = s’ k’. If k > k’ it is obvious from (ii), Lemma 4.2, that (J-)kf,,,k, = 0 for all s‘ 3 0. Thus,
+ +
+
~ [ Jf~ W ~I=. (pl~+ p 2 ) - k Q[(J+)v~.~ J-~,.~~I (PI
+
~
for all S, S’ 2 0,
=0
2 QCfs.0 ) ~ (J-)’fs,.k’] ~
and Q[fs,k,f s ’ , k ’ ] = 0 unless k = k’. Collecting all these facts together, we have the lemma. Among other things Lemma 4.3 proves that the vectors { f s , k ) are linearly independent. Hence, by our earlier remarks they form a basis for V2Q “yz and an analytic basis for bY2 Q GT2. V, @ V, can be decomposed into a direct sum of subspaces
where Vi is the subspace spanned by the basis vectors {fs,k), k = 0, 1, 2,... . T h e VL transforms under g(0, 1) according to the irreducible representation fW1+WZ-s,Pl+pz . This decomposition induces an analogous decomposition of bY2 Q a,.
We define Clebsch-Gordan coefficients H(p1 , I ; p, ,j I s, k), nonzero o n l y i f s k = I + j , by
+
c
s+k
fS.k
=
1=0
WPLI 1; ‘2, 9
S
+ k - 1 I s, A)
h,,,+k-,
,
A, S 3 0, (4.54)
+
i.e., f, t ( ~w, ) = xjLi H(pl, I ; p 2 , s k - 1 I s, k ) z ~ w ~ + A~ -simple ~ . consequence of this definition and (4.47) is the relation Q [ f s . , t h,,J , =
I! j!
W1,1; t ~ ,j 2 I S, h ) 7 sj,s+k-t he2
,
k, S, L i 3 0-
(4.55)
4-5. THE REPRESENTATION fo,,,,,
0To,, ,
We can use this relation to invert Eq. (4.54). Since "v, @ V2we can write
Fall
h1.j =
a=O
f
1 . j a.Z+i-a
9
99
is a basis for
1,i 2 0,
. An easy computation using Lemma 4.3 gives
for suitable constants
Comparing (4.55) and (4.56) we have
and finally
h,j = 1 '+j
s=o
H(pl, 1; p a ,j 1 s, I + j - s) I! j ! fs.l+i-s 4~iZMs. l+j-s
-
(4.57)
I t is easy to obtain a generating function for the Gkbsch-Gordan coefficients directly from (4.53). I n fact, for k = 0 and g = exp 6$+ in (4.53) one obtains the generating function
m s+1
=
1 "("
1=0 i=O
17'
P2)GH(pl
,j ; p2 ,s + I
-
j 1 s, I ) zjw8fZ-j. (4.58)
Equating coefficients of 6' on both sides of this expression we find
(I*lz -tP2w)x( z - w)' P1 + P2
s+ k
= 5=0
H(pl ,j ; p 2 , 5
+ k -j
I s, k) ziWa+k-j
or
-
k! s! (pl
+
n k-n
s+n-i
PlP2 (-1) p 2 ) k : n! ( j - n)! (k - n)! (s -j
+ n)! , s + k -j 3 0, (4.59)
where the sum on the right-hand side is taken over all integral values of n such that the summand is defined. The Clebsch-Gordan coefficients are independent of w1 and w 2 .
100
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
These coefficients are closely related to the Jacobi polynomials. In fact,
.F(-k,
-s
- k +j ;j - k + 1; -pa/&) r(j - k 1)
+
(4.60)
where we adopt the convention (A. 5) on the right-hand side of this equation. If we apply the operator T(g),g E G(0, l), to both sides of Eq. (4.57) we obtain
The right-hand side of this expression can be expanded in terms of the basis vectors h l , k .Equating the coefficients of hl,kon both sides of the resulting expression we get
s=o
or
The reader can work out some of the special cases of this formula for himself.
4-6. A REALIZATION
OF R(w, m , , p)
101
The above analysis demonstrates in a particular case the close connection between the reduction of tensor products of group representations into irreducible parts and identities expressing products of special functions as sums of special functions. We could carry out the same analysis for the representations ,,J, 8J Y a l p a . However, this procedure would lead to no new results for the associated Laguerre functions, so we omit it. T h e method presented here for the reduction of a tensor product of representations works only if the tensor product can be decomposed into a direct sum of irreducible representations. In particular, representations of the form R(w1 9 m19 PI>(20 R(W2 9 m2 9 P.2) or Ll,, 8lw2,-, cannot be so decomposed and our procedure fails. However, when we consider unitary group representations in Section 4-18, we will be able to use the tools of functional analysis to obtain deeper results relating the properties of Laguerre functions to tensor products of group representations. 4-6
A Realization of R(w, rn, , p) by Type D’ Operators
I n the first four sections of this chapter the irreducible representations of 9(0,1) were realized on spaces of analytic functions of one and four complex variables. Now we will construct realizations on a space of two complex variables, x and y . In fact, we will study those realizations of irreducible representations of 9(0,1) such that the operators J*, J3, E take the form
a
73=5,
J*
= e*Y
(*axa -
where p is a nonzero complex constant. These are the type D’ operators defined in Section 2-7. (The constant q occurring in the expression for the type D’ operators can be set equal to zero without any loss of generality.) T o begin with we construct a realization of R(w, m, ,p) (0 Re m, < 1, p # 0, and m, w is not an integer). Thus, we look for nonzero functionsf,(x, y ) = Zm(x)emvsuch that
+
<
102
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
+
for all m E S = {m, n: n an integer}, where the differential operators J*, J3, E are given by (4.64). In terms of the functions Zm(x), relations (4.65) reduce to (i)
d (z - h) Zm(4
= PZrn+l(X)I
T h e complex constant w is clearly irrelevant as far as the study of the special functions 2 , is concerned, since we could remove it by relabeling Hence, without loss of generality we can the functions 2, = ZA+o. assume w = 0. Also, there is no loss of generality for special function theory if we set p = 1 . The solutions of Eqs. (4.66) are well known. As remarked in Section 2-7, (iii) is the parabolic cylinder equation and its solutions are the parabolic cylinder functions D,(x), see (A. 16). In fact, for all m E S the following choices for Z, satisfy (4.66):
(4.67) Since w = 0 we must have 0 < Re m, < 1; hence, the elements m of S are not integers. I t follows from the above discussion that if the functions 2, , m E S, are given by either (1) or (2) in (4.67), then the functions f m ( x , y ) = Zm(x)emg form a basis for a realization of the representation R(0, m, , 1) of g(0, 1). By Theorem 1.10 this representation of Y(0, 1) by Lie derivatives can be extended to a local multiplier representation of G(0, 1). Thus, if we denote by 62 the space of all entire analytic functions of x and y, the operators (4.64) will uniquely define a multiplier representation T of G(0, 1) on GT. We will compute the induced multiplier representation in the usual manner. Since J+ = ev( ajax - ax) it follows that [T(exp b % + ) f ] ( x , eV) = Y ( X , ev, exp b / + ) f [ x ( b ) , bE/Z1,
fEGK
where d db
- x(b) = e y @ ) , d db
d dby(b) = 0,
-Y ( X , ev, exp b f + ) = - $x(b) ev(%(x,
ey,
exp b y + )
4-6. A REALIZATION OF R(w, m , , p)
103
= x , y(0) = y , ~ ( x ew, , e) =
1. T h e solution is
with initial conditions x ( 0 )
where t = ev. Similarly we find [T(exp c $ - ) f ] ( x , t) = exp
-xt-’c )f(x
- ct-1, t),
2
As mentioned in Section 4-1, i f g E G(0, 1) has parameters (a,b, c, T) then g = (exp b$+)(exp cf-)(exp .f3)(exp
a&).
Thus, T(g)f
=
T(exp b y + ) T(exp c$-) T(exp T $ ~ )T(exp aE)f
for all f~ a. Direct computation gives
.f(x
+ bt - ct-l, eTt) (4.68)
a.
for all g E G(0, l), f E (Note that t = ev # 0.) According to Section 2-2 the matrix elements of the operators T(g) with respect to the basis vectors fnL(x,t ) = Zm(x)tmare given by the matrices A,,(g), Eq. (4.13) ( w = 0, p = 1). This follows from the fact that the functionsf,, satisfy condition ( B ) . Thus, we obtain the relations
valid for all g E G(0, l), or exp
(-
t2b2- xtb - x t - l c 2 2 ~
+ t-2c2 -)bc2 -
D,(x
+ bt - ct-’)
a
(-c)-zL(-l)(-bc) m+l
= I=--m
Dm+L(x)ti, if
Zm(x) = (-l)”-m~Dm(x). (4.70)
104
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
Various special cases of these relations are worth investigating. If we set c = 0, t = 1, and make use of the limits (4.16) we obtain exp
b2 - -)xb D,(x -
(-
4
(-b)' Dm+dX)* + b) = C 7
2
z=o
If x = 0 this relation becomes
1, Eq. (4.70) reduces to
For b = 0, t
=
and, for x
0, to
=
4 7 A Realization of Iw,, by Type D' Operators
To obtain a realization of the representation I,,, of 8(0,1) by type D' operators acting on CZ it is necessary to find nonzero functionsf,(x, y) = Zm(x)emv( m = --w k, k 2 0) such that
+
JYm
= mfm
Efm
= Pfm Co.1fm
where
a
J3=3j9
J'fm
9
=
(J+J-
= Pfm+l
t
- EJ3)fm ==
(
J-fm Pwfm
"
J* = efv f- - &px ,
ax
= (m
+
w)fm-1,
(4.71)
9
E
= p.
In terms of the functions Z,(x) these relations take the form (4.66) where, as remarked earlier, without generality we can assume w = 0, p = 1. Thus,
(4.72)
(-
d2 xe 1 p +- - - m) Zm(x) = 0,
4
2
4-7. A REALIZATION OF
tw+
105
m = 0, 1, 2, ... . We can use these relations to explicitly compute the functions Z,(x). In fact, the relation J-fo = 0 implies
which has the solution Zo(x)= c exp(-xX2/4), c E fZ'. Choose For m > 0 we define the function Z,(x) by
c =
1.
zm(x) emu = fm(x9 Y ) = (J+)"fo(x, Y ) = (J+)"(zO(x)).
A straightforward induction argument yields zm(x) = exp
cf)
am
Gexp
(-
X2
T),
m = 0,1,2
,....
(4.73)
At this point we have found functions f m ( x ,y ) = Zm(x)emv, m 2 0, such that J3fm = mfm J+fm = f m + l , J-fo = 0. 9
We can then proceed exactly as in the proof of Theorem 2.2 to show that these functions must actually satisfy all the relations (4.71) for w = 0, p = 1. Hence, the functions Z,(x) defined by (4.73) are the solutions of (4.72). The Z,n(x) are easily expressed in terms of parabolic cylinder functions or Hermite polynomials H,(x). In fact, Zm(x) = (- 1)" D,(x) = (- 1)" exp( -x2/4) 2-m~2H,,,(2-1~2x),
m
= 0, 1, 2
,....
(4.74)
According to the above discussion the functions fm(x, y ) = Zm(x)emv, m 2 0,form a basis for a realization of the representation . As usual this representation of B(0, 1) can be extended to a multiplier representation T of G(0, 1) on a. The action of the operators T(g) has already been determined and is given by (4.68). T h e matrix elements of this representation with respect to the analytic basis {f,} are the functions B,,(g) computed in Section 4-2 ( w = 0, p = 1). Thus, we have m
[T(g)f,l(x, t ) =
1 B&) z=o
fi(x, t ) ,
K
= 0,1, 21.-,
or exp
(-
t2b2
-
xtb - xt-lc 2
~
2
+t-v
-
-)bc2
D,(x
+ bt - ct-1)
t'(
(4.75)
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
106
where t = eu. This last relation is valid for all b, c, x, t E fi?such that t # 0. I n terms of the Hermite polynomials H,,,(x)
=
exp(x2/2)2m/2Dm(21/2~),m
= 0,
1,2, ...,
Eq. (4.75) takes the form exp( - t W - 2xtb)Hk(x+ bt - ct-1)tk If c
=
0, t
=
m
=
1
(-bc)H,(x)tl.
(-c)k-lLi*-l)
(4.76)
1=0
1, this equation reduces to
and in the special case K
=
0, b = - y , it becomes (4.77)
which is a well-known generating function for the Hermite polynomials. Finally, when b = 0, t = - 1, Eq. (4.76) reads
We could find a realization of the representation I,,, using type D' operators in exact analogy with the procedure for tw,,. However, this construction will be omitted since the analysis leads to no new information about special functions. 4-8
Transformations of Type
C'
Operators
T h e type C' operators classified in Section 2-7 take the form
a
Js=6,
J i = e i . ( + - - - - - -a+ + P )l , a
ax
xay
4
E=p
(4.78)
x
where q, p E @, p # 0. (Without loss of generality, the constant p in the expression for the type C' operators can be set equal to 0.) These operators generate a complex Lie algebra isomorphic to g(0, 1). I n analogy with Section 4-6 we try to find a realization of the representation R(w, m, , p), where now the operators J*, J3, E have the form (4.78).
4-8. TRANSFORMATIONS OF TYPE C' OPERATORS
Thus, we look for nonzero functions fm(x, y ) {m, + k: k an integer}, such that JYm J-fm
= (m
= mfm
+
9
u)f-1*
Efm
,
= pfm Co,,fm
2
J+fm
(J'J-
=
107
Zm(x)emu, m E S =
= /tfm+l,
- EJ3)fm
= pufm
(4.79)
where the differential operators are given by (4.78). As was shown in Section 2-7, the functions Z,(x) must satisfy the equation
= pwZm(x),
111 E
s.
T h e solutions of this equation are called the functions of the paraboloid of revolution, ErdClyi et al. [l], Vol. 11, p. 126. I n particular, one such solution is (x2/4)' exp(-x2/8)
Pi(5 - 7
+ 4 ;25 + 1; x2/4)
+ +
where 4 = &(m - q) and 77 = $ p ( m q 2w integer there is a linearly independent solution (x2/4)-6exp( -x2/8) IFl(-5 - 7
+ 1). If m - q is not an
+ 4 ;- 25 + 1;xa/4).
Expressed in terms of generalized Laguerre functions, there are linearly independent solutions (1) (xa/4)' exp(--x2/8)Lf!i-+(xa/4) (2) (x2/4)4exp( -x2/8) Li;!5)+(x2/4).
and (4.80)
If we wish to study directly the properties of the confluent hypergeometric functions (or the generalized Laguerre functions), however, it is more convenient if we can find differential operators whose eigenfunctions are of the form
We shall transform the type C' operators into a new set of differential operators to bring about this desirable situation. To begin with we assume w = 0, p = 1. (As far as special function theory is concerned this is no loss of generality.) Then, the first solution (4.80) is f,,l(u, t ) = u-q/2e-U/2L (m-q) (u)tm where the variables u, t are defined by u = x2/4, t = eux/2. Denote by F the space of all functions
108
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
analytic in a neighborhood of the point (uo, 20) = (1, 1) and define the transformation cp of 9onto F by [cpfl(u,t) = u-Q/2e--u/2f(u, t ) , f E S. We can consider the mapping cp to be multiplication by the function y(u, t) = u-QJ2e--uJ2. This mapping is invertible with inverse cp-' defined by [cp-'fl(u, t) = u+Q/2e+u/2f(u, t). Under cp-l the basis function fm(u, t) defined above is mapped into [cp-lfm](u,t) = Lim-Q)(u)tm, which is in the desired form (4.81). If J is a linear differential operator on 9define the linear differential operator p on F by p(cp-If) = ( ~ p - ~ J c p ) ( < p - I f) = q+(J f) for all f E 9, i.e., p = (p-lJv.As a consequence of this definition we have
Properties (2) and (3) are trivial. Q.E.D. Thus the map J-. Jv' is a Lie algebra isomorphism. If
a a I =fW),, +fZ(%t),
+f3(%t),
fl I f 2 , f 9 E S ,
an explicit computation gives
(4.82)
where ~ ( ut), = u-Q/2e--u/2, y-l(u, t) = uQ/2eU/2. (Although we have made a special choice for q ~ ,Lemma 4.4 and Eq. (4.82) are clearly valid for any nonzero function y in F.)
4-9. TYPE C' REALIZATIONS OF R(w, m , , p)
109
Armed with these facts, we can compute the operators (J3)v, (J*)(P, (E)q where J3, J*, E are given by (4.78) ( p = 1). In terms of the coordinates u, t (u = x2/4, t = eYx/2), the results are (J')"
a
= t -at,
(E)V = 1 ,
As an example we verify the computation of (J+)v. Under the change of variables from (x, y) to (u, t), J+ becomes
Moreover,
Therefore,
T h e other relations are proved similarly. 49
Type C' Realizations of R(w, m , , p)
-
By construction the differential operators (4.83) generate a Lie algebra isomorphic to 9(0,1). These operators can be used to find a realization of the representation R(w, m, , p ) on F.Thus, we look for nonzero functions fnl(u,t) = Zm(u)tmdefined for all m E S = {m, k, k an integer}, such that relations (4.79) are valid. Here the differential operators are given by (4.83) and the superscript cp has been omitted. (There is no loss of generality for special function theory if we set o = 0, p = 1.) In terms of the functions Z,,(u) these relations are
+
110
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
for all m E S. The solutions of these equations are generalized Laguerre functions or, what is the same thing, confluent hypergeometric functions. Indeed, for all m E S the choice
zm( u ) = (-l)m-m,L(m-q) Q
(4
(4.85)
satisfies (4.84), see (A. 13). (It is left to the reader to find additional solutions.) Since w = 0 we must have 0 < Re m, < 1, so the elements m of S are not integers. I n the special case where m, - q is an integer the functions (4.85) have been obtained as matrix elements of representations R(o,m, , p). However, when m, - q is not an integer these functions do not occur as matrix elements. The above remarks have established the fact that the functions fm(u, t) = Z,,(u)t", m E S, form a basis for a realization of the representation R(0, m,, 1) of 3 ( 0 , 1) if the Z,(u) satisfy (4.84). I n the usual manner this realization can be extended to a local multiplier representation T of G(0, 1) on the space .F. According to Theorem 1.10 and relations (4.83), the local multiplier representation takes the form
for f E .F. If g E G(0, 1) has parameters ( a , b, c,
T),
then
T(g) = T(exp by+) T(exp c j - ) T(exp ~ $ 3 ) T(exp a&)
and [T(g) fI(u, t )
=
eaPbt(1 - c / t ) - q f [ ( u + b t ) ( l
-
c / t ) ,eT(t - c)]
(4.87)
for I c j t j < 1, ~ E The F .matrix elements of T(g)with respect to the analytic basis { f J u , t) = Zn,(u)tm},(4.85), are the functions A,,&) defined by Eq. (4.13), ( W = 0, p = 1). (The fact that this basis is analytic follows from (2.2) and the t dependence of the basis functions.) Thus,
4-10. TYPE C' REALIZATIONS OF
to,l
111
which simplifies to the identity
+ c/t)pL:)[(u + bt)(l + c / t ) ] = 1 c - l L ~ ~ ~ + , ( b c ) L : p + l ) ( u ) t l , I c / t I < 1, p , q E @, p + q not an integer. (4.89) a3
e-bt(l
Z=-m
Using the limits (4.16) we can investigate two special cases of this identity: If c = 0, t = 1, there follows L r ) ( u + b ) = eb
(-b)l
OD
-L:"fl)(u),
I!
1=0
while if b = 0, t = 1, there follows m
(1
+ c)pLy[u(l + c)] = 1
Ic I < 1.
(PTQ) c'L';-yu),
1-0
410 Type
C'
Realizations of
To,l
T o obtain a realization of the representation
to,lof 8(0,1) by operators
acting on .F we must find nonzero functions fm(u, t) m = 0, 1, 2,..., such that
=
Z,,,(u)tm,
-
Efm
J Y m = mfm
J-fm = mfm-1
v
fm
Co.1fm
==
J+fm
(J'J-
=fm+lp
- EJ3)fm
(4.90)
=0
for all m 2 0. (We assumef-, = 0.) Conditions (4.90) will be satisfied if and only if the special functions Z2n(u),m 2 0, satisfy the equations d (z - 1) &&) (-u
= Zm+1(4,
dZ
- + (q - m du2
(-u-&
-
I
d
-m
+ 41
= m&n-,(u),
+ u)- q ) ~ , ( u ) = 0. du d
(4.91)
(Here, Z-,(u) = 0.) Equations (4.91) determine the functions Z,, to within an arbitrary constant. Thus, the relation J-fo = 0 implies
112
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
+
q) Zo(u)= 0, which has the solution Zo(u) = cup, c an arbitrary constant. To normalize our solution we set c = 1. For m > 0 the functions ZnL(u)can be defined recursively by Zm(u)tm= fm(u,t ) = (J+)mfo(u,t ) = (J+)mZo(u). A straightforward induction argument yields ( - u d/du
m = 0 , 1 , 2,....
By definition the functionsf,(u, JYm =
mfm
t), m J'fm
i
(4.92)
> 0, satisfy the relations J-fo = 0.
=f m i l v
However, it follows immediately from the proof of Theorem 2.2 that thef,,, must actually satisfy all of the relations (4.90), and the functions Z,(u) must satisfy all the equations (4.91). The special functions 2, are conveniently expressed in terms of associated Laguerre polynomials I.:), m a positive integer. In fact,
,... .
Zm(u)= m! U ~ - ~ L : - ~ ' ( U )m , = 0, 1,2
(4.93)
This realization of 9(0,1) can be extended to a multiplier representation T of G(0, 1) on where the action of the operators T(g),g E G(0, 1) is given by (4.87). The matrix elements of T(g)with respect to the analytic basis {f,} are the functions B,,(g) studied in Section 4-2 (w = 0, p = 1). Thus, W
[T(g)fk](u, t ) =
1 Bzk(g)fz(u, t ) ! z=o
k
= 0,
1,2,---*
which leads to the identities k ! e-bt(1
+ b t / ~ ) q - ~ L ; - ~ ) [+u (bt/u)(l l - c/t)]tk (4.94)
valid for all b, c, t, u , q E @ such that I btju I < 1. Making use of the limits (4.29) we can derive some simple consequences of (4.94). For c = 0 , u = 5,
and, setting k = 0 in this expression we obtain a well-known generating function for the associated Laguerre polynomials:
+b ) = ~ C bW J - I ) ( u ) , m
e-bu(1
z=o
qE
c,
I b 1 < 1.
4-11. THE GROUP S,
When b
=
0, u
= t,
113
Eq. (4.94) becomes Ljp’(u - c )
=
c -L‘9+”(u). I! IC
1=o
cl
k--l
We could also use the type C‘ operators to find a realization of the representation lo,r. However, the identities for Laguerre polynomials obtained from such a realization are just (4.94) again, so this analysis will be omitted. In Chapter 5 we will derive additional identities for the Laguerre functions by relating these functions to the representation theory of sZ(2).
411 The Group 5, S, is the real 4-parameter Lie group of matrices 1 4e-%Z
<
+
where w = x iy E fZ‘, 0 multiplication law for S, is R{W,a,
a> * g@’,
= g lw
+i6 - weUj8 (4.95)
0 01
< 27r, (mod 27r), and 6 is real. The group
a’,8‘)
+ e-iw’, a + a’,6 + 6’ + s2 (iik.o’e-ia - w5’efiu)/.
(4.96)
Thus, g{O,O, 0} is the identity and the inverse of a group element is given by g-’{w, a,6) = g{ -e%, -a, -6). (4.97) As a basis for the real Lie algebra 0
-i/2
0
Y4of S , we can choose the elements
0
0
(4.98)
0
with commutation relations
K
= 1,2,3.
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
114
T h e 3 x 3 matrices
y*
=
%+, ,f-,
Tf2
8,
f3,
+ iYl,
2 3
=i
8
f3,
=
-i9,
must then satisfy the commutation relations [/3,
y*1 = hy*,
[Y+, /-I
=
-8,
[B,/'I
=
[E, 3 3 1 = 0.
Clearly, the complex Lie algebra generated by the basis elements (4.98) is 9(0,1). Thus 9(0,1) is the complexification of yk and 9, is a real form of 9(0,1) (see Section 3-5). Due to this relationship between the two Lie algebras, the abstract irreducible representations R(o,m, , p), I,,, , and Lw,, of 9(0,1) induce irreducible representations of .u?, . We shall investigate which of these irreducible representations of Y4can be extended to an irreducible unitary representation of S, on a Hilbert space. The technique for carrying out this investigation was discussed in Section 3-6 and applied to the real Lie group E3 in Section 3-7.
4-12
Induced Representations of Y4
As in Section 3-6 we consider a unitary irreducible representation U of S4on a Hilbert space X and define the infinitesimal operators J1 , J 2 , J3 9 Q by
for all f E B. Here, B is a dense subspace of X satisfying properties (3.45) and (3.46). On 9 these operators obey the commutation relations
[JI J21 = tQ, 9
[J3 Ji1 = I2 9
[J3 Jz]
=
-Ji
9
[Jk
k
3
Ql = 0,
=
1,2,3,
follows immediately from (4.99). Moreover, the operators J*, J3, E defined by J* = J2 + iJ1, J3 = iJ, , E = -iQ (4.101)
as
satisfy the commutation relations
[J+, J-I
=
-E,
[J3, J*]
=
iJ*, [J*, El
hence, these operators determine a representation algebra 9(0,1) on 9.
=
p
[J3, El
= 0;
of the complex Lie
4-12. INDUCED REPRESENTATIONS OF 9,
115
We shall first investigate which of the representations R(w, m, , p) of 9(0,1) can be induced by p on some dense subspace 9'of 9.According
to Lemma 3.1 we must have
k = 1,2939 -
f, h )
=
can write these conditions in the form
<J+f,hh) =
<J3f,hh>=
,
<Ef,hh)
=
(f, Eh)
(4.102)
for all f, h E 9.T h e representation R(w, m, ,p) is determined by the relations JYm J+fm
= mfm
=dm+l
=d m
Efm
9
J-fm
(m
1
+
<
+
9
(4.103)
w)fm-i
where p # 0,O Re m, < 1, m = m, k and k runs over the integers. We will assume that the basis vectors fn, are in 3 and use conditions (4.102) to find restrictions on w , p, and m, . Thus, M
= (JYm
,fn>
=
<
fm > JYn)
= n
9
fn),
(4-104)
or (6- n) ( fm ,fn) = 0 for all m, n in the spectrum of J3. Setting n = m, we can conclude that m is real, hence I m m, = 0. If m # n we have
,fm>
= <J+fm-I
,fm>
= (fm-19
J-fm>
+
=
(m
+
wI(fm-1
jfrn-1)
(4.105)
implies 0 < If, lz/\fm-l l2 = ( m w ) / p for all m E S. Since this last condition can never be satisfied for all m E S it follows that none of the irreducible representations R(w, m, ,p) of 9(0,1) are induced by unitary representations of S , . However, we shall have better luck with the representations tw,, of 9 ( 0 , l), determined by conditions (4.103) where now p # 0, o is arbitrary, and m = - w k where k runs over the nonnegative integers. As in the preceding case, in order that the operators (4.103) be obtained from a unitary representation of S, we must require (i) 0. From (3.46) we can show U(exp ay.)f m = exp( -ia J3))fm = cfmY,,,
+
+
116
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
for all m in the spectrum of J3. Since exp 27rY3= exp 0 = e (the = 1 for all m in the specidentity element of s,),we must have rZrim trum of J3. Thus, m and w must be integers. If we define new basis vectors jk by
K = 0 , 1 , 2 ,..., it is a simple consequence of conditions (i) and (iv) that (jk,jk,) = b S,,,, for all k, k’ >, 0, where b is a positive constant which without loss of generality can be set equal to 1. We can draw the following conclusions from this analysis: Any irreducible representation t,,, of ”(0, 1) induced by a unitary representation U of S, must be such that w is an integer and p > 0. Furthermore, the vectors form an orthonormal basis for H and satisfy the relations
Gk}
for all k >, 0 (j-l= 0). Conversely, we will soon show that each such representation (4.106) induces a unitary irreducible representation U of S, on a Hilbert space. A similar analysis proves: If the irreducible representation of ”(0, 1) is induced by a unitary representation of S, then w is an integer, p > 0 and there is an orthonormal basis Gk}, k = 0, 1, 2,..,,for satisfying the relations
for all k >, 0 (j-l= 0). Conversely we will show that each such representation (4.107) induces a unitary irreducible representation of S, on a Hilbert space. From (4.95) and (4.98) it is easy to verify the relations expyY1 = g{iy, 0, 0}, exp xY2 = g{x, 0, 0}, exp a f 3 = g{O, a,0}, and exp 6 1 = g{O, 0, S} for x, y, a, 6 real. If w = x iy it follows from (4.96) that is g{w, a,S} = (exp xYz)(exp yY,)(exp aY3)(exp(S XY/4)9). Thus, uniquely determined by the elements Yl, Y 2 Y , 31 , of 9,and according to Eqs. (3.46), the operators U(g), g E S , are uniquely determined by the infinitesimal operators J1 , J2 , J3 , Q defined by (4.100). The irreducible representations (4.106) and (4.107) uniquely determine the unitary representations from which they are derived. Before proceding with the determination of the irreducible unitary representations of S, it is useful to examine a realization of the represen-
+
+
s,
4-13. THE HILBERT SPACE 9
117
tation to,l of .Y4which is of importance in quantum mechanics: the occupation number space formalism for bosons. Let iX? be a complex Hilbert space with orthonormal basis vectors 1 n), n = 0, I, 2, ... . On &? we consider the annihilation operator a and its adjoint, the creation operator a+, defined by a 1 n) =
1 n - l),
a+ I n) = (n
+ 1)1/2 1 n + l),
n = 0, 1,2,..., (4.108)
and linearity. The number-of-particles operator N is defined by N = a+a and has the property NIn)=nln),
n = 0 , 1 , 2 ,....
(4.109)
As the names of the operators suggest, the eigenstates I n) of N are interpreted as eigenstates of n bosons. The vacuum or no-particle state is 10). From their definitions it is easy to verify that the four linear operators a, a+,N,I, (I is the identity operator on X ) satisfy the commutation relations [N, a+] = a+,
[a+,a] = -1,
[N, a] = -a,
[a+,I] = [a,I] = [N, I] = 0.
(4.1 10)
Comparison of Eqs. (4.106) and (4.108)-(4.110) shows that the occupation number space formalism gives a realization of the representation to,l of 9(0,l), hence, of Y4. In Section 4-14 this representation will be extended to a unitary representation of S, .
4 1 3 The Hilbert Space .F Our aim is now to extend the representations (4.106), (4.107) of 9, to unitary irreducible representations of S, . For this purpose we first define the Hilbert space 9on which these representations will operate. S is the space of entire analytic functions f of the complex variable t such that
+,
where d ( ( t ) = w-1exp( -lit)& dy, t = x + and the domain of integration is the complex plane. Every function f E .F can be expanded uniquely in a power series f(t) = x z = O a n t nwhich converges everywhere. The inner product (., .) is
=
Jfow&(th
f,
E
.F.
(4.1 11)
118
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
It is easy to see that the functionsj,(t) = P / ( ~ z !n) = ~ /0, ~ ,1, 2, ..., form an orthonormal basis for 9. I n fact, introducing polar coordinates t = reie, we have 1
eih-n'NJ
dB
I
m
p+n'+l
exp( - y 2 ) dr.
0
(4.1 12)
For any two functions f(t) = C antn and h(t) =
c
C b,tn
in 5 we find
m
(f,h )
=
(4.113)
n! inbn
n=O
and (4.1 14)
From (4.114), an entire function f is an element of 9 if and only if m l2 < CL). Equation (4.112) shows that the vectors G,} are orthonormal while (4.114) shows that they are complete; hence, they form an orthonormal basis for 9. T h e space 9 was introduced by Segal [I] and studied in detail by Bargmann [3,4]. We mention here some of the special properties of this space.
z n s o n ! I a,
I. Define the function e b E 9 for some complex constant b by xz=o(6t),/n!. It follows from (4.1 13) that for any f E 9,
e,(t) = exp(6t) =
(eb,.f> =
m
1 bnan = f ( b ) ,
(ea
eb>
= eb(b) = exp(6b).
n=O
Thus, eb acts like a delta function. Use of the vectors eb will greatly simplify the computations to follow. 11. Define the norm of a vector f E 9 by llfll = ( f , f ) ' / 2 . From the Schwarz inequality, I f(b)l = I<eb,f > ~ II e b II IIf II = exp@b,Q)IIf II. Thus, iff, h E 9,then I f ( b ) - h(b)l exp(6b/2) 11 f - h 11 which shows that convergence in the norm of 9 implies pointwise convergence, uniform on any compact set in C. 111. Finally, we quote without proof a theorem from Bargmann [3] which will be needed to justify several of our computations. Let Rk be
<
<
4-14. THE UNITARY REPRESENTATION (A, I )
119
k-dimensional real Euclidean space and Cm be m-dimensional complex space. We are concerned with integrals of the form
where D is a measurable set in Rk and z = (zl ,..., zm)is a point in an open set 8 of Cm. Assume that, for every z in a neighborhood N(lz, - bi 1 < pi) of the point b in 8, f is analytic in z for every 7 , measurable in T, and If(z,~)l< v(T),
I zj - bj I < Pi
9
where 7 is summable over D . Then F(z) is analytic in N, and its partial derivatives are obtained by differentiating under the sign of integration, the resulting integrals being summable. I n the following computations we freely interchange summation and integration without explicitly constructing the appropriate ~ I ( T ) .
4-14
The Unitary Representation (A, I )
We will now verify that all of the representations To+ of g(0, 1) with an integer and p > 0 induce unitary irreducible representations of S, . T o find these representations of S , note that for w an integer the multiplier representation (4.19), induced by to,+, depends on the parameter T only in the form e‘:
w
[B(g)f](z) = efi(br+a) (e7)-f(eTz
+ eTc),
f E @,
(4.1 15)
where g E G(0, 1) has parameters ( a , b, c, T). (If w is not an integer (eT)-w is not a single-valued function of eT.) Thus, the operators B(g) define a multiplier representation. of the 4-parameter matrix group G(0, 1)‘ with elements ceT
0
0
(4.1 16) 1
The matrices (4.1 16) are obtained from expression (4.4) for the matrices
of G(0, 1) by eliminating the last row and column. G(0, 1)’ is not simply connected. I n fact, g’(a, b, c, T ) = g’(a, b, c, T 27~i).However, G(0, 1) and G(0, 1)’ are isomorphic as local Lie groups, i.e., they have isomorphic
+
Lie algebras.
120
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
Comparison of (4.95) and (4.1 16) shows that S, can be considered as the real subgroup of G(0, 1)' consisting of all matrices g'(a, b, c, T ) such that Re 7 = 0, b = -t, and Re(-bc/2 + a ) = 0. Thus, g{w, a,S}
= g(i8 - w6/8,-w/2,5/2, -;a)
where w E fZ', 0 Q OL < 27, (mod 27), and 6 is real. Since S, can be embedded as a subgroup of G(0, 1)' we can obtain a global multiplier representation for S, by restricting the operators B(g), (4.115), to the group elements g in S,: [B(g)f](z)
= exp[p(-wz/2'+
i8 - &/8)
+ i u ~ ~ ] f ( e -+~e-"G/2), z
(4.117)
defined for every entire analytic function f and every element g in S, with parameters {w, 01, S}. It will be convenient to write this expression in a slightly different form. T o emphasize the fact that p and w can assume only restricted values we set p = I > 0 and w = h where h is an integer. Further, we introduce the new complex variable t defined by t = 1'I2z and the polar coordinates reie = w/2. I n terms of these new variables (4.117) becomes
where f is an entire function of t. T h e generalized Lie derivatives of this multiplier representation of S , are readily computed:
J3
Thus,
=i
(A - t
-),dtd
Q
= il.
4-14. THE UNITARY REPRESENTATION (A, I )
121
Applying these operators to the basis functions j J t ) = n = 0, 1, 2,..., we have
JYn = (n - 4jn, J+jn = [1(n
+
Ej,, = lin
,
.
J-j,,= (In)'/,
P/(n!)1/2,
.
These relations are identical with the relations (4.106) derived in Section 4-12. Thus to define unitary representations of S, induced by tl,* we need only introduce a Hilbert space such that the functionsj, form an orthonormal basis. The action of the group on this Hilbert,space will be given formally by (4.118). As we saw in the last section the Hilbert space 9 has the required property. Using the preceding facts as motivation we now construct the unitary representation (A, Z) of S, on 9. Here, A is an integer and 1 > 0. Let g be an element of S, with parameters {w, a, S} = {2reis, a,S}. Define the linear operator U(g) on 9 by [U(g)f](t) = exp[-W2treie - lr2/2
+ i1S + ix~!]f(te-~~ + ll/%e-h-i@),
f~ S. (4.1 19)
If g, and g, are elements of S, with coordinates rl , O1 , a, ,al and 1, , 8, , a 2 ,a,, respectively, one readily finds the coordinates I, 8, a, 6, of g,g2 E S, to be given by
Theorem 4.2
PROOF (i) have
(A, Z) is a unitary representation of S ,
Using the inner product (4.111) on 9 we
Unitarity.
(u(g>f, U(g)h) = m-1
.
J expI-1112ire-ie
-
/1/2treie - lr21
.,f( te-iu
+ /l/2ye-M-ie)
.h(te-iu
+ 11/2ye-iu-i0 ) exp(-tt)dxdy,
t =x
+ +,
for all g E S, , f, h E 9. With the introduction of the new complex variable t' = x' + iy' = te-iu + /l/zre-i=-i@this integral simplifies to 7r-l
Ifv)
h(t') exp(-t't';) dx' dy' = (f,h ) .
122
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
(ii) Representation property. given above, we have
If g, ,g, E S, with coordinates as
[U(g,) U(g,)f](t) = exp[-ZWr,ei@i - Zr,2/2 + i16, (te-hi
+ iAaJ[U(g,)f]
+ z1/2rle-%-%)
= exp[-ZWre'@
- lr2/2 + i16
+ iAa]f(te-ia + 21/2re-'U-'e
)
= [U(g,gJfl(t)
for allfE S, where the variables reie, a,6 are defined by (4.120).
4-15
Q.E.D.
The Matrix Elements of (A, 1)
Given g E S, with coordinates rete, a, 6, we shall compute the matrix e1ements n, m = 0,1,Z---, un,m(g) = < i n U(g)jm>, 9
with respect to the orthonormal basis {in).I n fact we can easily derive a generating function for these matrix elements. T o do this note that
(4.122)
where the last equality follows from (4.121). By equating coefficients of powers of s and u we easily find expressions for the matrix elements themselves. T h e result is Un,,(g) = exp[ia(A - m)
+ i16 + i(n - m)O]
4-15. THE MATRIX ELEMENTS OF (A,/)
or,
Unam(g)= exp[ia(X - m)
123
+ iZ6 + i(n - m)q
- exp (-
+
where the functions L p ) , a, b integers, a b 2 0, are the associated Laguerre polynomials. This result can be used to derive a new generating function for the associated Laguerre polynomials. Choose 0 = 01 = 6 = 0, I = 1 and substitute (4.123) into (4.122) to obtain (4.124)
Lemma 4.5
The representation (A, I ) is irreducible.
PROOF The method of proof is completely analogous to that of Lemma 3.2. Assume the lemma is false. Then, there exists a proper closed subspace JId of 9such that U(g) f E J?Y for all g E S, ,f E .1Let . P be the self-adjoint projection operator on A. Since JId is a proper subspace of 9 it follows that P # 0, I. Moreover, U(g)P = PU(g) for all g E S, . Let g, be the element of S, with coordinates (0,a,0}, i.e., reie = 6 = 0. Then, U(gol)jn = eia(A-n)jn for all n 2 0, which implies U(gol) Pj, = eiu(A-n)Pjn . Thus, Pj, = a&, a, a constant. Since P2 = P we have a t = a, for all n >, 0; hence, a, = 0 or a, = 1. By hypothesis there exist integers no , n1 2 0 such that Pj,, = 0, Pjnl = jnl. Thus, un0&9 = (in, U(g) Pj7Ll> = = 0 for all g E S, . This is a contradiction since none of the matrix elements (4.123) is identically zero for all g E S, . Q.E.D. T h e generating function G”l(g; s, u ) and the matrix elements U,,,(g) are functions of the group parameters r, 0, a, 6; hence, they can be integrated over the group manifold with respect to a suitable measure. As is well known from the general theory of topological groups (Naimark [I]), there is a unique (up.to a constant) measure d(g) on S, with the property 9
J f k J 4 g ) = JfW 4 g )
(4.125)
for every integrable function f on S, and every go E S , . (The domain of integration is the entire group manifold S, .) This measure is called the. (right invariant) Haar measure. The suitably normalized Haar measure on S, is given by d(g) =
Y
dr dB da d6
16,+
124
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
where r, 8, a, 6 are the coordinates of g E S, . Thus,
The reader can verify directly that d(g) has the property (4.125). T h e constant ( 16rr3)-I has been chosen for purposes of normalization. We now compute the integral
I GA*l(g;
m
u, S) GA'*''(g; u', s') d(g) =
1
p.q.n,m=O
m)
(The superscripts A, I are used to denote the representation (A, 0.) T h e first inteBral can be evaluated directly if one makes use of the simple expression (4.122) for G(-)and the properties of the vector eb . T h e result of the straightforward integration is Y
6(1 - I') J GAJ(g; U,S) GA'*"(g;u',s') d(g) = 41
exp(tiu'
+ is').
Here a,, is the Kronecker delta while 6(I - 1') is the Dirac delta function which arises from the symbolic integration formula
Theorem 4.3
6(Z - 1')
U;::(g) UA:p'(g) d(g) = 7 6A.A,6p,naqem.
Theorem 4.3 gives orthogonality relations for the matrix elements of the unitary representations (A, I). If the matrix elements are expressed in terms of associated Laguerre polynomials by (4.123), these orthogonality relations reduce to the formula
+
valid for all integers m, n 2 0 and all integers k such that n k 2 0, m+k>O. The group property U(g,g,) = U(gl)U(g,) leads to an addition theorem
m
=
C
I
u(gz>im>
4-16. THE UNITARY REPRESENTATIONS (A, - I )
125
for the matrix elements of (A, 1). Substituting expressions (4.123) for the matrix elements and simplifying we obtain (feie)m-nLcm-n)(f2) n
exp[
ei(e,-B
1 2
)
+
where m, n >, 0 and reie = r,eiel r2eiea. This formula is, of course, only a special case of the addition theorem (4.28) for the matrix elements of
Tw,c.
We can use (4.127)to derive a simple integral formula for the product of two associated Laguerre polynomials by setting 8, = 0 and multiplying both sides of the equation by ei(p-)ea. Integrating with respect to 0, we obtain
+
where reie = rl r2ei@sand p is a nonnegative integer. Since the operators U(g) are unitary and the basis vectors jnhave norm one it follows that
I un.m(g)I = I ( j n , U ( g ) j m > I G 1, which implies
4-16
I L;m-*)(r2)( < (m!/n!)1/2
n,m 2 0 , gESq* exp(r2/2).
The Unitary Representations (A, -I)
In exact analogy with the work of Section (4-14) we can construct unitary irreducibIe representations (A, -I) of S, on the Hilbert space which are induced by the representations J A , l of 9(0,1) (A an integer, 1 > 0). Rather than repeat the motivation for this construction we merely present the results. The construction of the representations (A, -1) of S, on .F is very similar to that for (A, 0. Define the linear operator V(g), on 9by [ V ( g ) f ] ( t= ) exp[Pttre-ie - h2/2
.f ( t e i u
- /1/Zreiu+if3 ) I
where g E S, has coordinates reie, a, 8.
+ ia(A + I ) - iD] f E 9 ,
126
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
Theorem 4.4 (A, - 1 ) is a unitary representation of S, . The matrix elements of (A, Z) are given by vn.m(g) = ( i n 7 V(g)i,>,
g
E
s, -
In analogy with (4.121) and (4.122), we obtain the generating function
=
exp[icu(A
+ 1)
-
i16 - &lr2 + u&(s - / W e i e ) + /1/2sre-ie], (4.129)
with the result V,,,(g)
= exp[ior(A *
+ m + 1) - i16 + i(m - n)O]
exp( -1~~/2)(n!/m!)~~~( -/1~2~)m--nL~mlm-n)(l~2). (4.130)
Note that the matrix elements (4.123) and (4.130) have the same dependence. They differ only in their dependence on a, 6, and 8. Lemma 4.6
The representation (A, Z) is irreducible.
because of the integration over the variable 8. The infinitesimal operators of the representation (A, -Z) f z , f 3 , 9 of Y4are given by to the elements
J~ = i (A
+I +t
J+ = -J,
+ iJ, = /l/z
= iJ3 = -A
d
dt '
-1-t-,
corresponding
dt
Hence,
J3
Y
d dt
J- = Jz
E
=
+ iJ1= /1/2t, -'
tQ = -1
4-17. THE TENSOR PRODUCTS (A, I) @ (A’, 1‘)
127
and the action of these operators on the basis vectorsj,(t) = tn/(n!)lPis J8jn
= (-A
-1
J+jn = ( l n ) 1 ’ 2 j n - 1 ,
-.)in, Ejn = -lj n ? J-j,
= [l(n
+ 1)I1’’jn+1
for all n 2 0. These relations are identical with Eqs. (4.107) (w = A, I ) and show that the representation (A, -l) of S, induces the representation of 9 ( 0 , 1).
p =
4-17
The Tensor Products (A, I )
0(A’,
1’)
We begin a study of tensor products of representations of S, by considering the representation (A, 1) @ (A‘, 1’)where A, A‘ are integers and 1,Z’ > 0. This representation is defined on the space S2 909 of entire functions f(t, p ) of the complex variables t and p such that
+
iy, and the domain of where d [ ( t ) = d e x p ( - t i ) dx dy, i = x integration is @. Clearly 9, is a Hilbert space with inner product
T h e action of g E S, on f E S2is given by
where g has the coordinates I, 0, OL, 6. It is simple to check that the operators UAJ(g) @ UA’.’‘(g) define a unitary representation (A, I ) @ (A’, 1‘) of S, on S2. However, this representation is not irreducible. T h e following paragraphs will be devoted to decomposing (A, 1) 0(A’, 1’) into irreducible representations. Since S, is not a compact group, general theorems on locally compact groups tell us only that (A, 1) @ (A’, Z’) is unitarily equivalent to a direct integral of unitary irreducible representations of S, . However, we shall show that in fact it is unitarily equivalent to a direct sum of unitary irreducible representations of S , .
128
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
T h e vectors jm,n= t"p"/(m! n ! ) l l 2 ,m, n = 0, 1, 2, ..., form an orthonorma1 basis for S2.By means of a change of variable a new basis can be introduced in S2on which the action of (A, 2) @ (A', l') becomes more transparent. Define new variables u and o by (4.132)
These equations can be inverted to give f = (1%
p
=
+ Z'y, + I1/2u)/(I+
- 1'1/2u)/(1
(l'%
I')1/2.
Furthermore, by evaluating the appropriate Jacobian it is easy to show d5(t) 4 p ) = d t ( u ) d5(b).
(4.133)
A polynomial in t, p can be written as a polynomial in u, w, and conversely. Indeed, i f f € S2,then f(t, p ) E h(u, o), where t, p , u, and w are related by (4.132) and h is an entire function of u, o. Again the converse holds. From these remarks and (4.133) we see that the functions unwm/(m!n ! ) l / 2 ,n, m = 0,1, 2, ..., form an orthonormal basis for .F2 . We @ UA'.''(g)defined by (4.131) to the funcnow apply the operator U**'(g) tion h(u, o) = unk(v) E S2, where n is a nonnegative integer and k is an entire function of o. T h e result is [UAeL(g) @ UA'-l'(g)h](u,v)
+ 1')1/2 vieie - &(l + I')r2 + ia(h + A' - n) + iS(I + l')]unk (ve-ia + i ( 1 + Z')1/2 e-fe-is). (4.134)
= exp[-(I
*
Thus, the functions u"k(v) E S2for fixed n form a basis space for the irreducible representation ( A A' -n, 1 Z') (compare with (4.1 19)). This proves:
+
+
Theorem 4.6
(A, 1) @ (A', l')
n=o
@ (A
+ A'
-n, 1
+ 1').
To study this decomposition in greater detail we adopt a new notation. Set
-
unvm
( m ! n!)1/2'
n , m = 0 , 1 , 2 ,....
4-17. THE TENSOR PRODUCTS (A, I) @ (A', 1')
129
Here, the superscripts on h denote the representation to which the corresponding basis vectors belong. From the last theorem, both the collection of vectors Gm,n}and the vectors {h$+A'-n*'+l')}form orthonorma1 bases for S2.Thus the j basis vectors and the h basis vectors can be expressed as linear combinations of one another, the coefficients of the expansion being known as Clebsch-Gordan coefficients. We now compute these coefficients. Clearly, m
Thus,
(4.135) Define the Clebsch-Gordan coefficients K [ l , q; 1',s 1 n, m],zero unless q s=n m, by
+
+
K[I, q; Z', s I n, m]
=
>'.
( j q , 8h:+A'-n*Z+Z') ,
From (4.135) we obtain K[1, q; I', s I 11, ml
0
if m + n # q + s ! (m)!( m
+ n - q)! (q)! (1/1 ) + ' n-q11/ 2
(1
Z/l')rn+"
Here the summation is taken over all integer values of a such that the summand is defined. Clearly, the Clebsch-Gordan coefficients are independent of A, A' and depend only on the ratio 111' of 1 and l'. It is easy to prove from (4.136) that K[Z, q; ,'Z
s I n, m] = (-l)n
K[Z', s; I , q I n, m].
130
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
It is also easy to find generating functions for the coefficients K. From
(4.135),
Since the left-hand side of this expression is invariant under the simultaneous transpositions w tt p, t t)z, K[Z, q; Z',
1 n, m]
s
=
K[Z, n; l', m 1 q, s].
T h e decomposition of (A, 1) 0 (A', l') into irreducible parts as given by Theorem 4.6 can be considered as a special case of the decomposition of the representation 0to,,r,of 9(0,1) studied in Section 4-5. I n fact, if w1 = A, w 2 = A', pl = I, p1 = l', the Clebsch-Gordan coefficients H, Eq. (4.58) in Section 4-5, are related to the coefficients K by the equation
-
(-l)n+m-Qn!
(n
F(-m, q
+ m - q)! (1 + l'/l)m
-n
- m; q - m $. 1; -z'/1)
T(q - m
+ 1)
.
(4.138)
T h e H and K coefficients differ by multiplicative factors because they have been defined with respect to different basis vectors. I n analogy with Eq. (4.62), we can derive the formula
g E S, , (4.139)
expressing the product of two matrix elements as a sum of matrix elements. T h e sum over k is finite because the coefficients K are nonzero for only a finite number of values of k.
4-18. T H E REPRESENTATIONS (A, I ) @ (A’, -1’)
131
By definition the Clebsch-Gordan coefficients satisfy the unitarity conditions m
These conditions can be used to derive identities for the Jacobi polynomials. T h e method needed to decompose the tensor product (A, -I) @ (A’, 4’) (A, A‘ integers, 1, 1’ > 0) is almost identical to that for (A, 1) @ (A’, I’). We define the unitary representation (A, - I ) @(A’, -l’) of S, on 55, by means of operators V**-l(g)@ W’P-~’ (g),g E s,9 given by [V”-yg) 0VA’.-’’( g ) f I(4 P ) = exp[(ll/zt
.f ( t e i u
pre-ie) - t ( l + i‘jr2 + ia(~+ A’
+~
- eiu/1/+ei9,
+ 2)
-
+
i ( ~ I’>s~
1.
(4.141)
peia - erul’1/2yei9
Introducing the variables u, ZI defined by (4.132) we can easily decompose this representation into its irreducible parts. The result is Theorem 4.7
(A, -1) @ (A’, -1’)
C 0 ( A + A’ n=o co
g
+ n + 1, -1-
I’).
T h e Clebsch-Gordan coefficients for this decomposition are (.&A ‘
’ p+A’+n+l.-Z-t’))’ m
=
K[I,
Q;
l‘, s 1 n, m],
(4.142)
where the coefficients K are defined by (4.136).
4-18 The Representations (A, I) @ (A’, -/’) T h e problem of decomposing the unitary representation (A, 1)@ (A’, -l’) where A, A‘ are integers and I, I‘ > 0 is more difficult than the problems considered in the last section. It naturally divides into three cases: (1) 1 > Z‘, (2) 1’ > 1, and (3) I = Z’. I n each case we define the representation (A, 1) @ (A’, - l ’ ) on S, as follows: For each g E.S,,designate by UA.#(g)@ VA‘n-J‘(g) = W(g) the operator [ W ( g ) f ( ] tp, )
+ l’1/2pre-ie + 1 ’ ) f + ia(A + A’ + 1) + i(l - l’)S]
= exp[ -1Wreig
- $(I
.
f(te-iu
+ ll/Zye-i9-k ,pek - [‘l/Pyei.eiB+iu 1
(4.143)
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
132
for all f E S2.It is easy to verify that these operators yield a unitary representation of S, on S2. CASE 1. I > 1'. We could treat this case by Lie algebraic methods in analogy with the work of Section 4-5. Indeed, we could compute the possible eigenvectors fo in S2of the operator J3 = -A - A' - 1 + t a / a t - p a/ap with the property J-fo = 0, where J- = P I 2 ajat W 2 p .For each such fo we could then show that the vectors (J+)"f0,n = 0,1, 2, ..., form a basis for an irreducible representation of S , . Here, Jf = PI2t I'll2 a/ap. Rather than work out the details of this construction it will be more convenient to present the results and verify that they are correct. We define the function t , P complex, by (4% u, t , p ) ,
+
+
G(u, a, t , p )
=
exp[(ut + vp)(l - 1'/Z)lI2
+ (uv - fp)(1'/1)1/2](1 - 1'/1)112
(4.144)
G(-)is a generating function for the vectors h p ) E .Z2which are defined by (4.144). Clearly, G(.) is an entire analytic function of each of its variables. Also it is square integrable with respect to either of the measures d((u) d((v) or d&t) d&p) taken over the domain C2. It will be shown that the vectors h p ) form an orthonormal basis for S2, and the matrix elements of W(g) with respect to this basis will be computed. T o this end we evaluate the inner product (G(u,v , *, .), [W(g)l G(w,z, - 7 --
I
.D' ( h p ) ,W(g) hLm))',
g E S4
.
(4.145)
The integration is carried out over the variables p , t . The inner product on the left-hand side of (4.145) can be evaluated directly by making use of the definitions (4.143), (4.144). This integration is elementary and can be performed with no other tool than the delta function property of the vectors eb . The result is
(4.146)
4-18. THE REPRESENTATIONS (A, I )
0(A',
-/')
133
where g E S, has coordinates I, 8, a, 8. We can draw several immediate conclusions from this equation. Lemma 4.7
(hin),W(g) hhm))' = 0 unless m
Lemma 4.8
For fixed n,
=
n.
(4.147)
PROOF Equate the coefficients of ( W Z ) on ~ both sides of (4.146). Comparing the generating function (4.147) with (4.122) we see that the vectors hin), k = 0, 1, 2, ..., form an orthonormal basis for the irreducible representation ( A A' n 1, I - )'Z of S4. Furthermore, i f g is the identity element in S4 (Y = 8 = a = 6 = 0), then according to Eq. (4.146) the vectors {hin)},n, k = 0, 1, 2,..., form an orthonormal set. We must still show that the {hin)}form a basis for Sz. T o do this we evaluate the inner product
+ + +
*7
-9
t,p)>'
u,v,
t,p € @.
This inner product can be computed directly by using the vectors The final result is exp(zit
+~
p =)
1 h',n'(u,v ) hjlR)(t,p). w-
eb
.
(4.148)
k.n=O
We have proved only pointwise convergence in (4.148), but convergence in the mean ( d [ ( t )d [ ( p ) ) follows from the orthonormality of the hin'. Set eu,Jt,p) = exp(zZt C p ) . From I, Section 4-13, we find (eu,o, f)' = f(u, v ) for all f~ S2. Thus,
+
m
f ( u , v ) = (e U . V
,f>'=
C
k.n=O
@P),.f>'hp)(u,v ) ,
proving that the h p ) form an orthonormal basis for S2.
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
134
If 1 > I'
Theorem 4.8
> 0 then
I) 0(A', 4')
(A,
2 @ (A + A' + n + 1 , l - l'). m
n=O
T h e basis vectors h p ) are easily computed from the generating function G( .): hjc")(t,p )
exp[ - ( t / / ) l / z t p ]
=
1(k!n!)1/*(t/ly/2(1 - I'/l)[(K+n+l)/z-aI a
tk-aPn-a
a ! (k - a)! (n - a ) ! '
k , n = 0 , 1 , 2 ,....
T h e summation is carried out over all values of a such that the summand is defined. Using this expression we can calculate the Clebsch-Gordan coefficients
where jp,s(t, p)
=
tQps/(q!s!)'/*, q, s 2 0. T h e result is
G [ l ,4;l', s 1 n, kl 0 -
s+k#q+n
if (9
+
tl
- k)! K ! n! (z'/i)*-k
1 ; 112
(q
(- l)Q-k(- r p ( 1 - l'/l)-a k a ) ! u! (k - a)!(n - a)!
-
+
if s + k = q + n .
(4.149)
From (4.144)one readily deduces the generating function exp[(ut
+ vp)(l
- f'/i)112
+ (uv - tp)(l'/l)llZ](l- l'/1)llz (4.150)
This function is invariant under the simultaneous transpositions t and u t)z, which implies
p
t)
G[l,4; 1', s I n,k] = G[l,S; l', q I k, n]
for all q, s, n, K 2 0. Another symmetry is u t)I, G[I,4; l', s I n, k]
=
z,
HP, which implies
G[2, k; l', 12 I S, q].
Comparing the generating functions (4.137) and (4.150) we find G[I,4;I', s I t l , k] = (1 - 1'/l)'I2 K[1 - l', 4; f ' , n I S, k].
4-18. THE REPRESENTATIONS (A, I ) @ (A', -1')
135
As a simple consequence of the definition of Clebsch-Gordan coefficients one obtains the formula
expressing the product of two matrix elements as a sum of matrix elements. In terms of the associated Laguerre polynomials these relations become
F(-m, -n; k
-
n
+ 1; -I'/(l
r(k - n
.L(q-s-m+n, m-n+k [(I
+ 1)
1'))F(-q,
~
--s;
k -s
+ 1 ; -I'/(Z
q k -s + 1)
- l')Y'].
- I'))
(4.151)
CASE 2. 1' > 1. In this case the functions hLn)are no longer elements of S2(since Z'/l > 1) and we must modify our procedure. I n terms of our Lie algebraic method we would look for eigenvectors f, of J3 such that J+f, = 0 and use these 'vectors to generate a new orthonormal basis for F2. However, we omit this and merely verify the validity of the results. The functions (t, p ) E 2F2, n, k = 0, 1,2, ..., are defined by means of the generating functions F(u, o, t, p ) :
fr)
F(u, v , t , p )
=
exp[(up
= exp[-
+ vt)(l
-
1/1')1/2
$(I' - Z ) Y ~ + im(X
+ (uv
-
tp)(l/I')1/2](1 - 1/1')1/2
+ A' + 1) - i S ( f ' - I)
136
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
( fp’,W(g)
Lemma 4.9
f?’)‘ = 0 unless m
= n.
Comparing this expression with the generating function (4.129), we see that the vectors { fp’}, k = 0, 1, 2, ..., form an orthonormal basis for the representation ( A A‘ - n, -(Z’ - I)) of S, . Finally, in analogy with case 1 we can show
+
where the convergence is in the mean. This is the completeness relation for the set { fp):”’) and proves: Theorem 4.9 (A,
If I’
> I > 0 then
I) @ (x’, -1‘)
1 @ (A + x’ co
g
71,
-(l’ - I)).
n=O
The Clebsch-Gordan coefficients for this decomposition are given by
,fF)>‘= G[l,s; l‘, q I n,4
<j**8
where G[*] is defined by (4.149). Note the interchange of s and q in the above expression. 4 1 9 The Representations (A, I ) @ (A’, - I ) According to (4.143) the representation (A, -I) @ (A’, --I) follows: Let g E S, with coordinates I, 8, a, 8. Then, [ W ( g ) f ] ( t , p= ) exp[11/2(pre-ie - treie) - 1r2
.
f(te-i.
+ y/l/2e-f@-iu
acts on T2as
+ ia(X + A’ + I ) ]
,peh - y/l/zeie+iu )
(4.152)
for every f~ F2. This representation of S, is independent of the parameter 6 ; hence, every element of the normal subgroup D of S, ,
D
= {g E
S, : g
= exp 6 9 , 6 real},
4-19. THE REPRESENTATIONS (A, I ) @ (A’, - I )
137
is mapped into the identity operator on P2.Thus, this representation can be considered as a representation of the (global) factor group S,/D g E, . In fact, if g, ,g2 E S , have coordinates y l , 8, , a1 and yZ, 8, , aZ , respectively, then the element gig,E S, has coordinates Y, 8, OL given by re’e
= r,e%
+ yzei(%*i),
+ a2
=
OL
(we ignore the parameter 6). For tp = -a these relations become = y,e%
+ y.#%+cpi),
+ Vz
V = Vl
9
which are identical with expression (3.43) giving the coordinate transformation for the product of two group elements in E, . From (4.152) it is easy to show that the infinitesimal operators J k , K = 1, 2, 3, and Q of the representation (A, l ) @ (A’, --I) are given by ip12
j1 = - -( t
2
a + p + -), a +at aP
a
p/2
= -( - - t
J2
-t-
2
a
a
J3=i(h+A’+1- ‘ z + P % ) ,
at
a + P - -)> aP
Q=O
with commutation relations
[Ji Jzl I
= 0,
[J3
7
Ji1 = J2
3
[J3
9
121
=
-Ji
Thus the infinitesimal operators generate a real Lie algebra isomorphic to Cf3. According to the above remarks, (A, I ) @ ( A ’ , --I) can be considered as a unitary representation of E, . From the general theory of representations of locally compact groups we know that this representation can be decomposed into a direct integral of irreducible representations of E3 (Naimark [l], Chapter 8). I n terms of special function theory this would yield an expression for the- product of two associated Laguerre polynomials as an integral over Bessel functions. We will determine this decomposition explicitly. Our realization of (A, I ) @(A’, --I) on the Hilbert space 2Ez is not a convenient one to use for this decomposition. However, there is another realization which is more convenient for tnis purpose. Bargmann has studied 9and S2in connection with the relationship between the usual representation of the canonical commutation relations in quantum mechanics and the Fock representation (Bargmann [3]). In the course of this study, he established a useful unitary mapping A of the Hilbert space L2(R)onto 9. The Lz(R)is the space of all complex functions on
138
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
the real line, square integrable with respect to Lebesgue measure, and with inner product
Since A(t,.) EL,(R) for each t E @, the above integral is always defined. T o understand the origin of the operator A and to verify its unitarity, we compare (4.154) with the generating function (4.77) for the Hermite polynomials. Clearly, m
In
m
(4.155)
where
It is elementary to verify the integral formula
On the other hand, using property 111, Section 4-13, we have
Thus,
These are the orthogonality relations for the Hermite polynomials, and they show that the functions {qn},n = 0, 1, 2, ..., form an orthonormal set in L,(R). I n fact, it is well known that the {p),} form an orthonormal basis for L,(R). We shall not demonstrate this here since group theoretic methods do not appear to simplify the proof. However, see Courant and Hilbert [l], and Bargmann [3].
4-19. THE REPRESENTATIONS (A, I )
0(A',
+
-I)
139
+
If + E L , ( R ) , then can be written uniquely in the form, = , c, E @, where convergence is in the mean. From (4.153) and (4.155) we have
z-o c,~,
f = A$
c cnjn m
=
E9
n=O
where the Gn}form an orthonormal basis for 9. Thus,
where the first norm is in 9while the second is in L,(R). Therefore, the linear mapping A is one-to-one, onto, and norm preserving, i.e., A is a unitary mapping from L,(R) onto 9. By construction we have jn = A v n , t f = 0 , 1 , 2 ,..., so the inverse map A-l of 9 onto L,(R) must satisfy the relations A-Yn . This suggests that the unitary mapping A-l is given by the integral
yn =
(A-YM
=
J.#AKt q ) f ( t )4 t )
(4.157)
for all j~ 9. However, this integral is not properly defined because A(., q) is not an element of 9for q E R. (Recall d&t) = r-l exp(-tf) dx dy where t = x iy.) To get around this difficulty note that
+
I 4 4 Q)I
< c exp(l t 12/2)
for all t E @ where C is a positive constant depending on q. If 0 < p < 1 we have I A(& q)] C exp[(p2 I t 12)/2].The A(@, q) is square integrable with respect to the measure d( for all q E R; hence, A ( p t , q) is in 9. Indeed direct computation (using property 111, Section 4-13) yields
<
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
140
Equating these expressions we obtain
0
< p < 1,
(4.158)
which is equivalent to Mehler's formula. (This is a special case of a generating function for Hermite functions which will be derived in Chapter 9.) Returning to the problem of justifying (4.157) we see that if f= c,& E . F and 0 < p < 1 then $,, EL,(R)where
x:Lo
W
+p(q)
AW, q ) f ( t )d t ( t ) =
=
1 p"cnvn(q).
n=O
If we define $ E L,(R) by $ = A-l f = C cnyn we have
II #
- #p
112
m
=
1 I cn 12(1 -
n=O
< II # (I2-
A simple limit argument shows
$,, -+ $ in the mean as p --f 1. Therefore, the unitary map A-l of 9 onto L,(R) can be defined by the expression so
(4.159)
for all f E .F. Clearly, f
=
A# if and only if $ = A-'f for $ EL,(R),
f E 9 .
Using the unitary maps A and A-' we can define operators TAs2(g), TA*-'(g), on L,(R) such that TA.Z(g) = A-1UA.t(g)A,
T**-'(g)= A-'V**-'MA.
T h e operators TA**'(g)for all g E S, define unitary representations of S, on L,(R) unitarily equivalent to the representations (A, *Z) of S, on F. These operators can be computed explicitly. If [z] is the element of S, with parameters (Y, 8, OL, 6) = (I,8, 0, 0), where z = x iy = 2Z1/*1eie, then [T**."zI+l(d= l.A-'UA*z[zlA#l(~). (4.160)
+
4-19. THE REPRESENTATIONS (A, I) @ (A', - I )
141
Following Bargmann [3], we set
f Then,
= A$,
fi
= UA*'[~]f,
T"."[z]$=
fdt) = Jrn exp[--tz/2 - z5/81A(t + 4 2 , q ) $( 4, !I) -03
and from the identity exp[-tz/2 - 22/81 A(t
+ 5/2, q) = exp[$iy(-d/2q
f 4211 A(t, p - X
/ d )
Thus,
for all t,4 EL,(R).Similarly, if (6) is the element of S, with coordinates ( r , 8, 01, 6) = (0, 0, 0, 6), we easily obtain
Let (a)be the element of S, with coordinates (0, 0, a, 0). Then,
Ife-& - 1 then TA.l(a) = I, while if e-im = - 1 we have [T** 2(a)$](4)= +(-q). However, if e-ia # fl it follows from Fubini's theorem that [TA-."(a)+](q) = eieA lim *+m
1"-,,
4'
u(e-'", q', q)
(4.163)
where u(e-im, q', q ) =
J A(t,)A(e-&t, q') de(t).
An explicit computation yields u(e-im,q', q ) =
e-ir(n/4-8/2)
( 2 I ~sin a
exp [i (qz + 4") 2
cot a
-
g-.
(4.164)
142
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
+
Here, OL = 2k7r $3, k an integer, E = k l , 0 < /I< m. (See Bargmann [3] for details. Equation (4.164) is presented for completeness. It will not be needed in the proofs of the various identities for special functions to follow.) When a! = r / 2 , TA-I(.rr/2)$is the Fourier transform of $. In a similar manner we can compute the operators T*--I(g)on L,(R), induced by the operators VA--l(g)on 9. T h e results are
[T"-'(WI(q) = e - " W q ) ,
[TA*-'(a)#](q) = e'e(A+l)lim n+w
(4.165)
1" -n
++t",
q', 9) $Q') 4'
for all fi €Lz(R),where the function u is given by (4.164). From the relation VA:k(g)= (jn , UA-l(g)jm) = (vn,TA*l(g)cpm) and Eq. (4.161), we obtain the identities n (44) exp(x2/8)(x/2)"-" Lcm-n)
=
1
+
00
[7~1/~2(n+~)/~n!]-l exp[-q2 - q x / d / Z ] H&) Hm(q x / l / Z ) dq, -00
(4.166)
exp(-y2/8)( ~ / 2 ) " Lhm"-")(y2/4) -~ - [7T1/22(n+m)/2n!]-1 Jm -m
exp[-q2 - @Y/d21K&)
4
where m, n are nonnegative integers and x, y >, 0. Equation (4.152) defines the representation (A, 1) @ (A', - I ) acting on 9, .F @ 9.We shall find it more convenient to consider this representation of S, as acting on the Hilbert space L,(R2)E L,(R) @L,(R). T h e appropriate transformation is easily carried out through the use of Eqs. (4.161)-(4.165). T h e elements of L,(R2) are complex functions square integrable with respect to Lebesgue measure in the real plane R2.TheLz(R2) is a Hilbert space with inner product
T h e operators T"'(g) @ T"p'(g) = A-' @ A-'(UA.'(g)@ V"*-'(g))A @ A
= M(g)
4-19. THE REPRESENTATIONS (A. I ) @ (A’, - I )
143
are defined on L2(R2)as follows:
z=x+iy,
for all Y E L , ( R ~For ) . notational brevity we have suppressed the dependence of M on A, A’, 1. T h e operators M(g) generate a unitary representation of S, such that M(6) = I for all (6) E S, . As remarked above this representation can be considered as a unitary representation of E, . As such it can be decomposed into a direct integral of irreducible unitary representations (p) of E, . We will carry out this decomposition. Given Y E L ~ ( Rwe~ define ) the function @ E L ~ ( Rby~ ) @(sl
, s2) = Y
-s1
+
s2
-s1
- s2
,
sl, s 2 €
Introducing the change of coordinates s1 = -(ql (ql - q2)/1/2, we obtain
R.
+ q2)/1/2, s2 =
Note that dql dq2 = ds, ds, . T h e operator M[z] acting on operator, which we will also call M[z], acting on @, [M[z]@](s,, s2)
Let the function
= eisi@(sl
, s2
(4.168)
Y induces an
+ x).
(4.1 69)
6 be defined by
6(s1 , u2) = (27~)-1/1
m -m
e-iu*s4(s1 , s2) ds, n
,
u2E
R.
(4.170)
(To be more precise we should write lim m,,Jin place of the integral in (4.170). However, this will be assumed to be understood in the following paragraphs.)
144
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
It is easy to show that M[z] acting on 0 induces an operator, which will also be called M[z], acting on 6 and given by [ M [ z ] ~ ] (,su2) , = e i ( s i ~ + u 2 x ) 6 ( s ,1 u,).
(4.171)
We now pass to polar coordinates and set x = T cos
y = T sin 0,
0,
where from (4.160) we have Define & by
=
T
&(P, Y) =
T h e action of M[x] induced on
s1 = p sin y ,
u, = p cos y ,
2 P h . Note that ds, du,
= p dp dy.
%Jsin Y,P cos Y).
6 then becomes
[M[z]&](p,y )
= eirp
(4.172)
coe(~-8@(p,y )
where & is square integrable with respect to the measure p dp dy. A comparison of Eqs. (4.172) and (3.56)gives the motivation for the above manipulations. We collect our results and use them to compute the matrix element (!PI , M[z]Y2), Yl, Y2€L,(R2).T h e result is (by use of the Plancherel theorem for the Fourier transform)
where the & j , j = 1, 2, are defined by
Using the Fourier integral theorem, we can invert (4.174): Yj(ql , q2) = (2n)-’I2
Iw exp[iu,(q, -W
- q2)/1/2]
rq
, u2) du,
.
(4.175)
Finally, a straightforward computation gives
(!PI , M ( a ) Y 2 )= exp[-ia(h
+ A’ + I ) ]
Comparing these results with the formulas for the irreducible representations ( p) of E, derived in Section 3-8, we obtain Theorem 4.10
(A, I )
0(A’,
-I)
@
I (PIP W
dp.
145
4-19. THE REPRESENTATIONS (A, I ) @ (A', - I )
Theorem 4.10 is merely a symbolic way of denoting the relationship between (A, I ) @(A', -1) and ( p ) given by Eqs. (4.173) and (4.176). Explicitly, we can use our decomposition to relate matrix elements for the representations on the left- and right-hand sides of (4.173). T h e vectors Yn,m(ql , q2) = Fn(ql) Vm(q2),0 d a, m < 00, form an orthonorma1 basis for L2(R2).From (4.174) the transformed vectors &,,,(p, y), peiy = u2 is, , are given by
+
where the last equality follows from (4.161) and a change of variable. Equation (4.175) then yields the relation
between associated Laguerre polynomials and Hermite polynomials. T h e matrix elements of M[z] acting on L,(R2)are given by
, M[zI Ymsj>= q:;([4) yy(".I>.
On the other hand, from (4.173), (4.177), and (3.57), we have
(4.178)
146
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
where m, j, n , k , are nonnegative integers. This equation is the desired identity between associated Laguerre polynomials and Bessel functions whose existence is suggested by Theorem 4.10. As a special case of (4.178) set K = j = 0 to obtain
420 The Representations (p) @ (A, I) As we have seen, the representation (p), p > 0, of E, on the Hilbert space H,constructed in Section 3-8, can be considered as a representation of S, . Thus, it makes sense to define a representation (p) @ (A, 1) of S4on the Hilbert space H @ 9, We will decompose (p) @ (A, 1) into a direct sum of irreducible representations. The elements of H @ 9 are formal series
such that
c m
m
n=-m k=O
I :C l2 < 00;
2, c: E
@, 0
< 297,
(mod 297).
Clearly f ( y , .) E 9for almost every y and f(., t) E H for fixed t . In general f ( y , t) does not converge pointwise. H @ 9 is a Hilbert space with inner product
Thus, if f ( y , t)
=
cteinytk/(k!)1/2, h(y, t)
bk,einytk/(k!)1/2,we have
c c$$ m
=
m
n=-m k=O
’
0, form an The vectors fn,k(y,t) = einYtk/(k!)ll2, n, k integers, k orthonormal basis for H @ 9. (For a careful treatment of tensor products of Hilbert spaces see Murray and von Neumann El].) Corre-
4-20. THE REPRESENTATIONS (p) @(A, I )
147
sponding to everyg E S4with coordinates ( I , 8,a, 6 ) we define the operator T(P)(g)@ UA,l(g)= N(g) on 3? 69 as follows: [N(g)f](y,
t) =
+ i d + i16 + ipr cos(8 - y ) ] f ( y + a, te+ + l1I2re-i(O+a)),
exp[--ill%efe - h2/2
f E 2P 09, (4.179) where h is an integer and p, 1 > 0. I t is a straightforward computation to check that the operators N(g) yield a unitary representation (p) 6 (A, I ) of S , . T h e infinitesimal operators J k , K = 1,2, 3, and Q corresponding to this representation are
Q
= il.
These operators satisfy the commutation relations
[Ji JJ 9
=
$0,
[J3
3
J11
=
J2
9
[J,
9
J21
= -Ji
9
[Jk
k
9
Q] = 0,
=
1,2,3,
so they generate a Lie algebra isomorphic to yk . As in Section 4-5, we can use the operators J+ = -J,
a . + iJ1 = 11Pt - iee-iy, J- = + J z + iJ1= 11P- + ?&Y 2 at 2 a . a J3=iJ3=--h+t-++-, E=-iQ=l at ay
to decompose 3? @ S into a direct sum of subspaces, each subspace
irreducible under the action of N(g). Rather than repeat this analysis, however, we will merely present the results and verify that they yield the desired decomposition. Let n be an integer, u E fZ’, and N(n, u; y , t ) the function N(n, u; y , t ) = exp[-p2/81
+ iny + ut + ~pe+/21l/~- t ~ e ~ y / 2 P / ~ ] (4.180)
N( -) is a generating function for the vectors hLk)E X @ .F defined by (4.180). Clearly, N(.) is an entire function of the variables u, t and is an element of 3? @ .F for all values of n and u.
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
148
We will show that the vectors hAk),k = 0, I, 2,..., n = 0, f l , A 2,..., form an orthonormal basis for M Q 9 and will calculate the matrix elements of N(g) with respect to this basis. T o do this we compute the inner product (in X Q 9)
where the integration is taken over the variables y, t . (The validity of the expansion on the right-hand side of (4.181) follows from property 111, Section 4-13.) The integral on the left-hand side of this expression can be evaluated directly using the delta function property of the vector eb and the fact that (zT)-1 ei(m-n)ydy = m.n * 0
The result of the integration is 8m,nexp[ia(A
+ n) + ilS
-
lr2/2
+ ue-iu(s + 11/2re-ie)- sP/2~eie] (4.182)
The following are immediate consequences of (4.182): Lemma 4.11
If m # n, then (h%),N(g)h;?) = 0 for all g E S,
Lemma 4.12 For fixed n the vectors {hik’},k = 0, 1, 2, ..., form an orthonormal basis for the irreducible representation (A n, I ) of S, .
+
PROOF If m = n, Eq. (4.182) is identical with the generating function for the matrix elements of the representation (A n, I ) . Equation (4.182) shows that the totality of vectors {hik’},k = 0, 1,2, ..., n = 0, f l , f 2 , ..., form an orthonormal set in M @ 9. We will sketch the proof that they span M Q 9. Suppose the vectors {hik)}do not span. Then, there exists f E S? @ S, f # 0, such that (f,hkk)) = 0,k, fn = 0, 1, 2,... . From (4.180)
+
0=
“ 9
c; .hf> *I
for all integers n and complex numbers u. Using the delta function property of the vectors eb where b = u - pe-i~/2P/~, we obtain
o _= [
27
- 0
f (7, u
-
-)pe-iv 2111’
[
exp -in7
1-upe‘y
-t
21’12
’
dy.
(4.183)
4-20.THE REPRESENTATIONS (p) @ (A, I )
149
This equation forces us to conclude that
for all y, u ; hence, f = 0. This contradiction proves that the vectors hkk)form an orthonormal basis for 3? @ 9.
5 m
Theorem 4.11
( p ) @ (A,
I)
n=-m
@ (A
+ n, I).
T h e basis vectors hkk) are easily computed from the generating function (4.180): h r ) ( y ,t )
=
(k!)-'/"t
+ (pe-*y)/211/2)kexp[-p2/8Z + iny
-
(tpeiy)/2W2]
(4.184)
where th last summation is taken over all integer values of such that the summand is defined. The Clebsch-Gordan coefficients for the decomposition given by Theorem 4.1 1 are obtained directly from (4.184). Recall that the vectors fW,*(y, t) = e i w W / ( q ! ) 1 / 2 ,q, &w = 0,1, 2,..., form an orthonormal basis for X @ 9. The Clebsch-Gordan coefficients E(w, q ; n, R ; p 2 / l ) are given by E(w, q; n, k P 2 / 4 =
, y))
if w + k = n + q .
(4.185)
Here q, R are nonnegative integers and n, w are integers. From (4.180) there follow the generating function N(n, u ; y , t ) = exp[--p2/8Z
+ iny + ut +
- tpei~)/2P/2]
(4.186)
150
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
and the symmetry relations
E(w,q; n,k;p2/1) = E(w
+ m - n,q; m,k;p2/Z) = E(w - n,q; 0,k. / '" ')'
E(w, 9; 0,k;p2/1) = (-l)"+"E( -w, k;0,q; p2/1).
(4.187)
Comparison of (4.186) with the generating function (4.124) for associated Laguerre polynomials yields ( q ! / k ! ) 1 /exp 2
(-
-$)
(p/21112)"qLbk-+")(p/211/2)= E(q - k,q; 0, k;p2/1). (4.188)
I n terms of matrix elements with respect to the two sets of basis vectors { f w , , ) and {h$), the decomposition given by Theorem 4.11 becomes
1 E(w, q ; n, n + n=-m m
u:;(g) ui:i'(g) =
*
E(s, k;n, n
+k -
- w ;p2/1) S; p2/1) u ~ ~ ~ ! n + k - - s ((4.189) g)-
Here, w and s are integers while q and k are nonnegative integers. The matrix elements Ug,)s(g)are expressed in terms of Bessel functions by Eq. (3.57). Making use of (4.188) we can reduce (4.189) to the identity exP(f2)(r)s-w
J w J W Jy-Q'(t.2)
where the sum is taken over all integers n such that n 2 s - k and n 2 w - q. If q = w = s = k = 0 this formula simplifies to the wellknown relation
The decomposition of the representation ( p ) @(A, -I) of S, is very similar to that presented above and leads to no new relations for special functions. We quote only Theorem 4.12
(p)
0 (A, --1)
C
n=-m
@(A
+ n, -I).
421 A Contraction of g(O.1) In Chapter 2 it was shown that the Lie algebra g(0,O) E F3@ (8) was a contraction of g(0, 1). Here, we shall use this fact to obtain relations between associated Laguerre polynomials and Bessel functions.
4-21. A CONTRACTION OF 59(0,1)
151
The infinitesimal operators J k , k = 1, 2, 3, Q corresponding to the representation (A, I ) of S, on 9are given by
")dt
J3=i(A-t
'
Q=d
and obey the commutation relations
We relabel the orthonormal basis vectors jk(t) = tk/(k!)'12 of 9 as follows: f, = jA+,, n = -A, - A 1, - A 2,... . I n terms of this basis the matrix elements of the operator J3 are given by
+
(J;'),,,.
=
= --in an,,,
n, R' = -A,
k
+
-A
(4.192)
+ 1, -A + 2,... .
Moreover it follows from Chapter 3, Eq. (3.64), that the operators K, 2, 3, given by
= 1,
K,
=
-ipcosa,
K,
= -+sin
U,
,
a aa
K - -3
-
are infinitesimal operators induced by the representation ( p ) of E, on the Hilbert space H,and obey the commutation relations [K3,
Kl]
=
Kz ,
[K, , K2] = -Kl ,
[K, , K2] = 0.
(4.194)
152
4. CONFLUENT HYPERGEOMETRIC FUNCTIONS
Clearly, K, , K2 , K, generate a Lie algebra isomorphic to 8;.In terms of the orthonormal basis h,(or) = eina, r t n = 0, 1, 2,..., for X‘the matrix elements of these operators are
(4.195)
Here, the inner product (., .)* corresponds to the Hilbert space X . Choosing a parameter > 0 we define a new set of generators for the Lie algebra 9,. T h e new operators are Gi
=~
J
i
G2 = cJ2 ,
t
G3 = J3 ,
Go = EQ.
(4.196)
T h e structure constants for the commutation relations of the G operators are functions of E . I n fact, [G3 G11 9
=
G2,
[G3
7
‘321
=
-GI
[GI , G2]
9
=
+Go
9
[Gk Go] = 0, k = 1,2,3. 9
As E -+ 0 the structure constants approach limits which are the structure constants for a new 4-dimensional Lie algebra. In the limit we have [G3 ,GI]
=
[G3 G21
‘32
9
= --GI
9
[GI , G2]
= 0,
[G,, Go] = 0, k = 1, 2, 3.
This Lie algebra is isomorphic to 8,@ {Go],where the commutation relations for a basis of g3are given by (4.194) and {Go}is the 1-dimensional real Lie algebra generated by G o . Thus, 8, @(Go} is a contraction of 9, (see Section 2-5). We show how an irreducible representation of the contracted algebra 8, @ {Go}can be obtained from a sequence of representations of 9,. Consider the representation (A, p2) of S , (A an integer, p > 0). T h e matrix elements of the G operators for this representation can easily be computed from (4.193) and (4.196). We take the limit of these matrix elements as E -+ 0 and A -+ 00 in such a manner that r2A -+ 1. For GI the result is lim(Gf-*~p2)n,n~ = !-y ( f, , E Jlfn,)
+
r+O
ezA+l
= -
4 d L . d + &+i.n,)
= (K%.n*
(4.197)
4-21. A CONTRACTION OF 9(0,1)
153
(n and n' are held fixed). Similarly, we find
l j ~ ( G ~ a ~ p 2=) n0,. n ~ n,n' = 0,fl, A 2,... .
Thus, we have derived an irreducible representation of the contracted algebra 8.@ {Go}as a limit of irreducible representations of Y4. T h e effect of this limiting procedure on the matrix elements of the representation (A, p2) of the group S4is easily computed from the above considerations. For example, from (4.123) we know that the matrix elements of the unitary operator e--YJl = UA+"(g),y > 0 ( g has coordinates {reie, a ,A} = {--&iy, 0, 0}), are given by
Similarly from (3.57) we see that for the representation ( p ) of E , the matrix elements of e--YK1 are given by
(4.200)
which simplifies to the relation lim m-nynL2)(y2/m)= Jn(2y). m-m
(4.201)
This proof of the limit (4.201) is merely formal; we have not verified the validity of (4.200). However, now that this limit relation has been motivated it is easy to prove it directly from the power series expansions for Laguerre and Bessel functions. T h e preceding discussion was concerned with the relationship between matrix elements of unitary representations of S4and E, . The restriction to unitary representations is not essential, however, and similar arguments can be used to relate matrix elements of irreducible representations of G(0, I ) and matrix elements of irreducible representations of T3 . We omit this.
CHAPTER
5 Lie Theory and Hypergeometric Functions
With respect to a suitable basis, the matrix elements of irreLxi le representations of g(1,O) g $42) @ (8) can be expressed in terms of hypergeometric functions. In Sections 5- 1 to 5-7 these matrix elements will be computed and employed to derive identities and differential relations for the hypergeometric functions. Sections 5-8 to 5-15 are devoted to the realization of irreducible representations of 4 2 ) in terms of type A and type B differential operators. I t will be shown that the most general hypergeometric function can be obtained as a basis vector transforming under the type A operators. Similarly, the most general confluent hypergeometric (Laguerre) function will arise as a basis vector transforming under type B operators. This connection between type A and B operators and special functions will prove to be a powerful tool for deriving identities involving hypergeometric and Laguerre functions. U p to isomorphism the complex Lie algebra $42)has two distinct real forms: su(2) and L(G,). I n Sections 5-16 to 5-18 all of the unitary irreducible representations of the real Lie groups SU(2) and G3 will be obtained by restriction of corresponding local multiplier representations of SL(2).The implications of unitarity for special function theory will be briefly discussed. Finally, in Section 5-19 the Lie algebraic fact that F3 is a contraction of 4 2 ) will be used to derive a formula expressing Bessel functions as limits of Jacobi polynomials.
5-1. THE REPRESENTATION D ~ ( umo) ,
5-1
The Representation
155
m,)
Dp(u,
According to Theorem 2.3 the irreducible representation Dp(u, m,) of g(1,O) is defined by complex constants p, u, m, such that neither m, u or m, - u is an integer, and 0 Re m, < 1. The spectrum of this representation is the set S = {m, n: n an integer}. There is a basis {fm : m E S}for the representation space V such that
+
+
for all m
E
<
S. These operators satisfy the commutation relations
[J3, J*]
=
[J+, J-]
&J*,
= 2J3,
[J*, E]
=
[J3,
E]
= 0.
(5.1)
We will construct a realization of the algebraic representation Dp(u, m,) such that Jf, J3, E take the form of the differential operators (2.35) acting on a vector space of analytic functions of the complex variable z. Let Vl be the complex vector space of all finite linear combinations of the functions h,(x) = zn, n = 0, f l , *2,... . I n Eqs. (2.35) set h = m, , tg = -u - m, , to obtain
The basis vectorsf,, , m E S, are defined byf,(z) and all integers n. Thus,
+
J"fm= (m, z J+fm
=
"1
((m,- u)z
Efm
= pfm
m,
+n
+ n)zn = mfm,
zn = (m,
+ z2 dz d l zn = (m - u)znf1 = (m -
d (+ + z)
~ -= j - ~
= h,(z) for m =
Zn
+ u)z+l
=
(m
-
+
, (5-3) %I-1
1
*
Clearly, the operators (5.2) define a realization of Dp(u, m,) on "v; . We can now apply the procedure described in Section 2-2 to extend this realization of Dr(u, m,) on Vl to a local multiplier representation of
156
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
the local Lie group G( 1,O) =GL(2).However, since 9(1,O) s $42) 0(8) and p ( b ) = E is a multiple of the identity operator for every irreducible representation p of 9(1, 0) listed in Theorem 2.3, the nontrivial part of the representation theory of 9(1,O) is concerned solely with the action of p on 4 2 ) . We can set E = p = 0 without loss of generality for special function theory. It is clear, moreover, that the representation D ( u , m,) of 9(1, 0) induces an irreducible representation D(u, m,) of sZ(2). T h e action of D(u, m,) is obtained from Eqs. (5.1)-(5.3) by suppressing the operator E. In the remainder of this section we will study the representation D(u, m,) of 4 2 ) and extend it to a local multiplier representation of the local group SL(2). Similarly, rather than study the representations t", I:, D(2u) of 9(0,1) given by Theorem 2.3, we will restrict ourselves to the representations t,, I,, D(2u) of $42) obtained by setting E = p = 0. It was shown in Section 1-2 that sZ(2) is the Lie algebra of the local Lie group SL(2), SL(2) = lg
a b d ) : a , b, c, d , E @, detg = 1
= (c
i.e., the group of all 2 x 2 complex matrices with determinant equal to 1. Theorem 1.10 can be applied to extend the realization of D(u, m,) defined on V, to a local multiplier representation of SL(2) defined on (2, , where OT, is the complex vector space of all functions of z analytic in some neighborhood of the point z = 1. Clearly, OTl 3 V, and M, is invariant under the differential operators (5.2). Thus, these operators generate a Lie algebra of generalized Lie derivatives corresponding to a local multiplier representation A of SL(2) acting on QI, . T h e action of the 1-parameter subgroup {exp cy-,c E e'} of SL(2) on (2, is obtained by solving the equations
+
dz
-- - 1,
dc
d
"(9, exp c/-)
=
-
u
+ m,
v(z0, exp c f - ) Z
with initial conditions z(0) = zO # 0, "(9, e) = 1, where exp c f -
=
(-c
1 0 1).
T h e solution of these equations is z(c) = zo - c,
v(zo,exp c f - )
If j~ OT, is analytic in a neighborhood of [A(exp c / - ) f ] ( z o )
=
(1
=
(1 - C / Z ~ ) U + ~ O .
we have
- c / ~ ' ) u + m 0 f ( 9- c )
5-1. T H E REPRESENTATION D ~ ( um,) ,
157
and A(exp c f - ) f E Qdl if 1 cis? [ < 1 and s? - c is in the domain off. Similarly, 20 [A(exp b f + ) f ] ( . “ ) = (1 - bzO)u-mof
(-)1 - bzo
where A(exp by+) f E a, if domain of f;and
I b y I < 1 and P i ( l
-b
y ) is in the
[A(exp ~ / ~ ) f ] ( z “ )= emosf(eTzO)
where A(exp 7 f 3 ) f E Qdl if eTs? is in the domain off. If g E SL(2) and d # 0 it is a straightforward computation to show that g
=
(exp b’f+)(exp c’f-)(exp
T‘f3)
<
where b’ = -b/d, c‘ = -cd, = d-l, 0 Im 7’ < 4 ~ Thus, . for I b’ I, 1 c’ 1, 17’ 1 sufficiently small, the operator A(g) is given by
for f ad
-
E Qd, and z # 0 in the domain of f. We have used the fact bc = 1. T h e multiplier Y takes the form
v ( z , ~= ) (d
+
+
b ~ ) u - m o ( ~ C/Z)”+~O.
As it stands, the final expression in (5.4) is not well defined; it is not even a single-valued function of the group parameters. However, it is obvious that for g in a sufficiently small neighborhood of the identity element, v ( z , g ) has a unique Laurent series expansion in z. TOgive a precise definition of the operators A(g) we restrict g to the open set N c SL(2), N
= { g E SL(2) : I c/u
I < 1 < 1 d/b 1,
-7
< arg a , arg d < T}.
As a local Lie group, N is isomorphic to SL(2). Given f E CZl let Dfbe the domain of f, i.e., the open set in containing 1, on which f is defined and analytic. For every g E N, z E @ let zg be the complex number
c,
az
+c
zg = bz Id
and define 9,(g), the domain of g, by q ( g ) = (f.
uc, : Ig E D,},
1E
c.
158
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
Then, ifg E N , and f E 9,(g) we have A(g) f E 0ll where A(g)f is defined by the Laurent series expansion of v in (5.4). In particular, D*G)/ = (I cl. I
< I z I < I dlb I> n (2 : zg E 0,).
T o give a precise statement of the representation property of the operators A(g) we define the set N(g,), for every g o E N , by N(g,) = {g E N : gg, E N} and let ”(go) be the connected component of N(g,) containing the identity element e E SL(2).Then, ifg, E N , g , E “(g,) we have g,g, E N; and, for every f E a1 such that f E 9 , ( g Z ) and A(g,)f E g1(gl),we can conclude that f E g1(gz2)and “g,gz)fl(z)
=
[A(gdA(gz)f)l(z)
(5.5)
for all z in some nonzero neighborhood of z = 1. (The restriction of
g, to N’(g,) is needed to make the representation single-valued.)
Following the procedure of Section 2-2 we can restrict the multiplier to TI. (T, is the space of all finite linear representation A from combinations of functions of the form A(g) f where g E N, f E Vl .) Clearly is invariant under A. Furthermore, every h E 7,has a unique Lsurent expansion m
h(z) =
C
anzn,
n=-m
a n € @,
which converges absolutely in an open annulus Dh about z = 0, including the point z = 1 in its interior. Thus the basis functions f,(z) = h,(z) = zn, m = m, n, n an integer, form an analytic basis for Tl (see Section 2-2). are defined by The matrix elements A&) of the operators A(g) on
+
m
[ A ( g ) h J ( z )=
l=-m
Azk(g)hz(z),
g EN, k
= 0,
&2,*--, (5.6)
or
where I cia I From ( 5 . 9 ,
< I z I < I d/b 1,
-7r
< arg a, arg d < r , andad - bc = A(gl)[Ak2)hkl,
which leads to the addition theorem
= 1:
5-1. THE REPRESENTATION DI.(u, m,)
159
valid for all g, , g, E N such that g, E N'(g,). Expanding the left-hand side of (5.7) by means of the binomial theorem and computing the coefficient of z' we obtain an explicit expression for the matrix elements: =
+ +
us+w-%-'r(s k 1) F(- s - I, - t r(S 1 + 1)(k - I)!
+
+ k;k - 1 + 1; bc/ad) if k >, 1, (5.9)
if 1 k. Here g E N, r is the gamma function, F is the hypergeometric function defined by its power series expansion, (A.4), and s = u m,, t = u - m, . Note that neither s nor t is an integer. The two expressions (5.9) can be combined into one by noting that
+
F(-
s - I, - t
is defined even when K - I we obtain
+
k; k - 1 r ( k - 1 1)
+
+ 1; b c / ~ d )
+ 1 is a negative integer. In fact, using (AS)
(5.10) for all integers I, k. The matrix elements can be written in other equivalent forms by making use of the transformation formulas (A.8), but this will be left to the reader. It will sometimes be convenient to adopt a different coordinate system for N. Namely, we write
= (exp a$3)
(exp -w
("+
2
+
"-))
(exp BY3)-
A simple computation shows that the coordinates a, b, c, d = (1 are related to the coordinates a, /3, w by
+ bc)/a
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
160
T o make this change of variable one-to-one on N we would have to put some restrictions on the range of the complex variables a, /3, w. Thus, (a, /3, w ) and (a i?~,/3 - ir, - w ) correspond to the same coordinates u, b, c. Further the coordinate transformation is not defined for bc = 0, so the coordinates (a, /3, w ) cannot be extended over all of N. For our purposes,however, it is enough to note that if we can assign to g E N coordinates (a, /3, w ) satisfying (5.1 l), then for suitably restricted values of the parameters the matrix elements A,,(g) are given by
+
+
where cosh w = 2bc 1 and 231sm(z) is defined by (A.9), (iii). (Indeed (5.12) is obviously true for a, /3 real and w > 0. Its domain of validity can then be extended by analytic continuation.) Clearly, the branch points of B>'"(z) are located at z = -1, + I , 00. Thus, if the z-plane is cut along the real axis from - 00 to 1 this function can be analytically continued to the whole z-plane outside the cut. Furthermore, we have the relations 23yyz) = 2 y ( z ) , 2 3 y ( z ) = 23?E1(z). (5.13)
+
An important special case is b;*-r(z) = Sp(z)
where the generalized spherical harmonics 23: are defined by (A.9), (ii). T h e functions B;"(z) will be encountered frequently in the remainder of this chapter. If the expression (5.10) for the matrix elements is inserted into (5.7) one obtains the generating function (1
+
z-')S(l
+ bz)t =
c
I=-m
2-l
r(s
+ 1)
q s +I 1
+ 1)
F(- s - I, - i; - I q- I 1)
+
+ 1; b ) ,
< I z / < Ib-'I.
(5.14)
We can derive addition theorems for the hypergeometric functions by substituting (5.10) into (5.8). T h e result is (after some manipulation)
c 02
=
j=-m
u5
F(s, t + j - k;j
r(i+1)
+ 1; b ) F ( s - j ,
+ k + 1; c) + K + 1) ,
t ; -j
r(-j
(5.15)
5-2. THE REPRESENTATION
1%
161
where s, t are arbitrary complex numbers, not integers, and k is an integer, When k is a negative integer, the function [ r ( k ) ] - l F ( s ,t; k ; z) is defined by (AS). As a power series in z, F(s, t; k; x) converges absolutely for I z I < 1. However, if a cut is made in the complex plane from + I to +a along the real axis, this function can be analytically continued to a function analytic and single-valued in z throughout the cut plane. Thus, Eq. (5.15) can be analytically continued in the variables a, b, c to a larger domainof validity than that originally implied by the addition theorem(5.8). In fact, the left-hand side of (5.15) is defined and analytic for all a, b, c E @ such that I b I < I a I < I l/c I and(c b/a)(a I)( 1 b/a)-l(l ac)-l does not cross the cut. Hence, the right-hand side of (5.15) must also be defined and analytic for these values of the variables, where in addition we must require that b and c do not cross the cut. Two special cases of this formula are of interest. For b = 0,
+
(1
+ ac)-t(a +
while if c
=
F (s, t ; k 1)k
+ 1 ; -c ( a 1
r(k
+
+
+
= L I7-
+ 1)
3-0
0,
k
aj-k F(s, t
+j
j--w
5-2 The Representation
-
k;j
r(i+ 1)
+ 1; b) ,
I b/a I < 1.
t,
The irreducible representation t,”of 9(1, 0) is defined for all p, u E fZ’ such that 2u is not a nonnegative integer. T h e spectrum of this representation is S = {- u + n : n a nonnegative integer}, and the representation space V has a basis JYm J-fm
= -(m
(Here f-u-l
= mfm
+
9
U)fm-1
= 0, so J-f-,
Efm
= pfm
1
Ci ofm = (J+J= 0.)
{fm
J+fm
+
,m
E
S> such that
= (m - U )fm+ i
J3J3 - J3)fm
,
= u(u
+ 1)fm -
162
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
As in the last section we can construct a realization of 7,' on a space of analytic functions of one complex variable z such that J*, J3, E are the differential operators (2.35). Let V2be the complex vector space of all finite linear combinations of the functions h,(z) = zn, n = 0, 1,2,..., and define the basis functions f,, m E S, by f,(z) = h,(z) where m = -u n. I n Eqs. (2.35) set X = -u, c, = -224 to obtain the differential operators
+
Js = - u
d + z -d z ,
E
d dz '
J+ = -2uz + z 2 -
=p,
J-
= -
dz *
These operators and basis functions satisfy the relations
J+fm
= (-224%
J-fm
= -
d dz
+ z2 -)dzd
zn
zn
= (-2u
= -nzn-1 ==
-(
+ n)zn+l = (rn - u)
fm+l
, (5.16)
+rn)fm-1,
Efm = pfm
for all m E S and, thus, determine a realization of t,". I n the usual manner this realization can be extended to a multiplier representation of G(l, 0) defined on a; 3 V 2 .Here 12;is the complex vector space of all functions f analytic in some neighborhood of z = 0, i.e., the space of all functions f of the form
n-0
where the power series converges in a nonzero neighborhood of z = 0. However, as was shown in Section 5-1, without loss of generality we can set E = p = 0 in (5.16) and consider the representation of 4 2 ) induced by the representation 1,"of Y"( 1,O). Hence, we shall suppress the operator E in the following paragraphs and extend the realization of T, given by (5.16) to a local multiplier representation B of SL(2) on
r,
a;.
The multiplier representation B was computed in Section 1-4. Indeed, from Eq. (1.69)' we have (5.18)
5-2. THE REPRESENTATION
tU
163
where g E SL(2) is given by (1.15). However, to define this multiplier representation precisely we restrict g to the open set P C SL(2) where P = {g E SL(2) : d # 0, 1 arg d
1 < r}.
P is a local Lie group isomorphic to SL(2), i.e., P and SL(2) have the let D, C e( be the domain of f. same Lie algebra. For every f E Furthermore, if zg = (ax c)/(bz + d) for every g E P, x E (z: define 9 , ( g ) , the domain of g , by
+
q g )
= {f.
a; : og E D,},
0 E c?.
Then if g E P and f E 9 , ( g ) we have B ( g ) f E by the power series (5.18). I n fact, DB(~)f =
{I
I < I d/b 1)
(5.19)
where B(g) f is given
{ z : ZgEDt}.
Given go E P define the set P(g,) C P by P(g0) = {g E p :ggo E PI
and let P'(g,,) be the connected component of P(g,,) containing e (the identity element of SL(2).The fact that B is a local representation of P is expressed as follows: (1) If g , E P , g , E P'(g,), then g g , E P ; and (2) for every f E such that f E Q2(g2)and B(g,) f E 9,(g1), we can conclude that f E 9,(glg2) and
for all z in a nonzero neighborhood of z = 0. The restriction of g , to P'(g,) is required to make the representation single-valued (221 may not be an integer). However, if 2u happens to be a negative integer this restriction is not necessary.
According to our usual procedure we define the matrix elements
B,,(g) of the operators B(g) by
or (5.21)
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
164
where I arg d I < 7~ and ad - bc = 1. From the representation property of the operators B(g) we obtain the addition theorem
c m
BZk(glgZ)
=
i=O
BZdgl)Bjk(gZ),
2 O,
which holds for all g, ,g, E P such that g, E P'(gz). T h e matrix elements were computed explicitly in Section 1-4:
if l > k > O ,
(5.23)
However, from (A.5) it follows that both (5.22) and (5.23) retain their validity for all k, I >, 0 without restriction. The hypergeometric series in these expressions break off after a finite number of terms. Hence they are hypergeometric polynomials in bc/ad (the Jacobi polynomials). In terms of the variables (a,p, w), Eq. (5.1 l), the matrix elements are Blk(g)= p ( - u + l ) & ( - ~ + k )
where the functions Bi*m(z)are defined by (A.9). (See the remarks immediately following Eq. (5.12).) From (5.21) and (5.23) there follows the generating function
valid for I z 1 < 1. The addition theorem for the matrix elements B,, takes the form (a
+ b)yl + ac)ZU(a +
1)k-1
+ 1; (ac + b)(a + l)(a + b)-1(1 + r(k - 1 + 1) F ( 4 , - 2u - k + j ; j - 1 + 1; b) = 1 ai F(-l, -2u; k - I
m
i=o
..)-I)
r(j- 1 + 1)
F(-j, -2u; k - j + 1; c) , r(k - j 1)
+
lacI
(5.25)
165
5-3. THE REPRESENTATION Ju
valid for all nonnegative integers k, I, and all complex numbers u such that 2u is not a nonnegative integer. T h e function [r(n)]-lF(s,t; n; z ) is defined by (A.5) when n is a negative integer. If b = 0, Eq. (5.25) simplifies to (1
+ ac)2U(a + 1)k-l
F( -I,
-224; K
af F ( - j - I , -2u; K - j r(k - j -I 3.
=c’T. W
j-0
while if c
5-3
=
-
I
+ 1; c(a + 1)( 1 + ac)-1) + 1) I + 1; c ) , Iac1<1,
r(k - I -
+ 1)
0, it becomes
The Representation
1,
According to Theorem 2.3 the irreducible representation 1,” of Y(1,O) is defined for each p, u E f2 such that 224 is not a nonnegative integer. T h e spectrum of this representation is the set S = {u - n: n a nonnegative integer}. T h e representation space V has a basis {fnl , m E S} such that JYm I-fm
=
-(m
+
=
mfm
9
lofm-1,
Elm
= Pfm
C,,ofm
=
J+lm ==
9
(m - u ) f m + l ,
+ J3J3 -
(J’J-
J’))fm
=T
U(U
+ 1)fm.
(Here fu+l = 0.) As in the first two sections of this chapter we could find a realization of 1,”in terms of the differential operators (2.35). However, it is more convenient to use the operators J3 =
-z
d d z,
E =p,
J+
d
= -dz,
J-
=
-2 uz
+
22
d
-
dz
(5.26)
which are a modification of (2.35). These o2erators satisfy the commutation relations ( 5 . I), hence, they generate a Lie algebra isomorphic to Y(1, 0). We will realize 1: on the space V2of all finite linear combinations of the functions h,(z) = zn,n 2 0, where J*, J3, E are given by (5.26).
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
166
Thus, if the basis functions fm , m E S, for V2are defined by fm(x) where m = u - n, n 0, we have
>
J+fm
=
d (- z)
J-fm
=
(
Efm
= pfm *
zn =
-2~2
-nzn-l
+ z2 dzdl zn -
= =z
(m - u)fm+l
(--2~
,
+ n)z"+l
=
-(m
+
=
xn
(5.27) u)fm-1
9
which yields a realization of 1,". As usual we set E = p = 0 in (5.26), (5.27), and restrict ourselves to consideration of the representation 1, of 4 2 ) induced by 1:. T h e realization of 1, given by (5.27) can be extended to a local multiplier representation of SL(2) on the space G!li of all functions of the complex variable x, analytic in some neighborhood of z = 0. Since the procedures involved in deriving this multiplier representation and computing its matrix elements are so similar to the work of Section 5-2, only the results will be presented. Consider the neighborhood of U of the identity element of SL(2) defined by U
= {g E
SL(2) : a # 0, I arg a I
< w}
and the set g 3 ( g ) , %(g)
where
= {fE
a;
: b/a E
D,>,
D,C @ is the domain off. T h e action of the local Lie group U on
G!li is determined
by the operators C ( g ) where
(5.28)
defined for all f E 67;, g
E
U , and z E @such that
121
and
+a
dz +bEDt. cz
Given go E U, define the set U(go)by U(go) = {g E U :gg,, E U } and let U'(go)be the connected component of U(go)containing e. T h e fact that the operators C ( g ) form a local representation of U is expressed by the relation [C(g,g,)fl(z)
EC(g,)(C(g,)f
)1(4,
5-4. T H E REPRESENTATION 4 2 ~ )
167
valid for g2 E U , gl E u'(g2),f E 9 3 ( g 2 ) , C ( g 2 ) f E =93(gJ and z in a suitable nonzero neighborhood of z = 0. The matrix elements for this multiplier representation are defined by
c Cz,(g)h,(z), W
[C(g)h,l(z) =
2 0, g E
1=o
u,
(5.29)
which leads to (cz
+
+ b ) k = c Czk(g)Zl, W
(dz
u)PU--lC
1 cz/u I < 1.
1=0
(5.30)
A comparison of (5.21) and (5.30) shows that the matrix elements can be
obtained from those of the preceding section by making the interchanges a t-,d, b t-,c. Thus,
+ 1; bclad) + 1) - d%2"-WT(2u - k + 1) F( -k, - 224 + I; I - K + 1; bc/ad) . (5.31) r(l- k + 1) r ( 2 u - 1 + 1)
CZ-lCk)=
+ l)F(-Z,
d'&-"k-Zr(k r(l+1)
- 224 + k; k - I r(k -1
Finally, we have the addition theorem
valid for g, E U , g, E U'(gp).The identities for special functions obtained from (5.32) do not differ from (5.25).
5-4 The Representation D(2u) The finite-dimensional representation Dp(2u) of 9(1,0) is defined for all p E @' and all nonnegative integers 224. The spectrum S of this representation is given by S = {u, u - 1, ..., -u 1, -u} and the vectors fm , m E S, form a basis for the (224 1)-dimensional representation space V , where
+
+
JYm J-fm
=
(Here,
-(m f-u-l
= mfm
+
U)fm-l
3
Efm
= t+fm
Cl,ofm
= fu+l = 0.)
=
9
(J'J-
J'fm
+
= (m - u>fm+l9 J3J3
- J3))fm =
U(U
+ l)fm -
168
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
A realization of this representation can be constructed on the space V~ZL)ofalllinearcombinationsofthefunctionsh,(z) = zn, n = 0, 1,..., 224, such that the operators J*, J3, E take the form (2.35). Indeed, for A = - u, c3 = -224 in Eqs. (2.35) we have E=p,
J ~ = - u + z - ,d
J + = - ~ u z + z ~d -
dz '
dz
J-=-d
dz ' (5.33) Define the basis functions f,, m E S by fm(z) = h,(z) where m = - u + n, 0 n 2u. These operators and basis functions satisfy the relations
< <
+ n)zn = mfm ,
--u
J+fm = (-2uz J-f m = -
Efm
22
dzn dz
= pfm
+ d-)dz =
2n =
-nzn-1
(-224
= -(u
+ n)zn+l = ( m - u )fm+l ,
+ m).frn-1*
>
and define a realization of Dp(2u). As remarked earlier, without loss of generality for special function theory we can set E = p = 0 in (5.33) and restrict ourselves to consideration of the representation D(2u) of 4 2 ) induced by the representation DO(2u) of 9(1, 0). Thus we suppress the operator E. T h e representation D(2u) of $42) on YCu)can be extended to a multiplier representation of SL(2) on V u I)n. fact, the relevant computations were carried out in Section 1-4. It was shown there that the action of SL(2) is determined by operators D(g), such that (5.34)
for all z E fZ', f~ VU), and g E SL(2). Furthermore, D(g,gz)f
= D(g,)[D(g,lfl
for ail g1 > gz E W 2 ) .
In this case the operators D(g) are defined for all g E SL(2), not just in a neighborhood of the identity element. T h e matrix elements Dlk(g)of D(2u)with respect to the basis hk(z)= zk, K = 0, 1,..., 224, are defined by [D(g)h,l(z)
2u
=
1 D,,(g)h,(z),
1-0
0
(5.35)
169
5-4. THE REPRESENTATION D(2u)
or (bz
+ d)Z"-k(az +
c &(g)zL. 2u
c)k =
(5.36)
l=O
A simple computation yields
+ + - K + 1) F( -K, -2u + I ; E - K + 1 ; bc/ad) r(z- K + 1) r ( 2 u - E + 1) 0 < k , I < 214, +
a'd2u-kck-'r(K + 1) F( --I, -224 K ; k - 1 1; b c / ~ d ) q l + 1) r ( k - 1 1)
Dtk(g)
- akd2U-%-kr(2u -
9
(5.37)
where the hypergeometric polynomials are defined by (A.5) when k - I (or 1 - k) is a negative integer. From (5.36) and (5.37) follows the generating function
0
In terms of the variables become
D&)
< k < 2u.
p, m), Eq. (5.11), the matrix elements
(a,
(5.38)
= p(-u+')eB(-u+k)
where 2 3 ~ ~ "is defined by (A.9). (See the remarks following (5.12).) T h e addition theorem for the matrix elements is
Substituting(5.37) into this expression we obtain, after somesimplification, (a
+ b)"l +
ac)2U-k(a
+
+
1)k-L
+ +
----I++ (ac
-I, - 2 ~ k; k - 1 1; ( a r ( k - E 1)
b) (a b) (1
+ 1)
+4
+ j ; j - I + I ; b) F(-j, -2u + K ; k -j + 1 ; ~ ,) 1 aj F(--Z, -2ur(jr(K -j + 1) E + 1) 2u
=
j=O
valid for all integers I, K, 2u such that 224 2 1, k
(5.39)
0, and all a, b, c E @.
170
5-5
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
The Tensor Product D(2u)
0 D(2v)
We now turn to the problem of decomposing the tensor product of two finite-dimensional irreducible representations D(2u) and D(2v) of $42) into a direct sum of such representations. Although the solution to this problem is well known, it will be treated again here because of the simplicity of the method and because the treatment generalizes to tensor products of infinite-dimensional representations of sZ(2). Let D(2u), 2u a nonnegative integer, be a finite-dimensional representation of 4 2 ) defined in Section 5-4. Recall that this representation was ) by the functions hk(z)= zk, realized on the space V ( ugenerated K = 0, 1,..., 2u. However, we will find it more convenient to use instead of the vectors h,(z), the renormalized basis vectors
(-z)U+*
= [(u
+ n)!(u
'
- n)!]1/2
n = - u -,
u
+ I ,..., u.
(5.40)
I n terms of this new basis for V ( % the) action of the differential operators (5.33) becomes
J3Pp, = nP,,
J-P,
J+P, =
=
[(u - fi)(u
[(u - fi
+ + I)1"2Ppn+1
+ I)(u + n)]"'~,+ -
>
(5.41)
If we define the matrix elements QiIj(g), g E SL(2), by
we obtain
I! (2u - 1 ) ! ] 1 / 2 Dzk(g), k! ( 2 -~ k)!
0
< 1, k < 2%
(5.43)
where DIk(g)is given by (5.37). Moreover, it is easy to derive the generating function 1
-[(6z
(2u)!
+ d ) + w(az + c)I2" = C Pk-u(w)Q21~'(g)P1-u(z). 2U
(5.44)
Z,k=O
T h e representation D(2u) @ D(2v) can be defined on the vector space V ( u0 ) V(w consisting ) of all polynomialsf in the variables z , w such that 2u f ( Z p
2.21
w) =
aklzli'UIz?
k=O 1=0
akz E
@-
5-5. THE TENSOR PRODUCT D(2u) 0 D(2v)
171
T h e vectors p k , l , Pk,l(Z?w) =
( -z)"+k( -w)"+2 [ ( u + k)! ( v I ) ! ( u - k)! ( v - I)!]'/
+
-u
' (5.45)
-v
form a basis for clr(u) @ V ( vI)n. particular, the dimension of this space 1). T h e action of SL(2) is given by the operators is (224 + 1)(2v T(g),g E SL(2),where
+
IT(g)fl(z, W ) = (bz
aw + ~ ) ' V+Wd)'"f ( z + r d ' c d ) az + c
f c
(5.46)
for all f E V ( u@ ) V ( wClearly, ). the operators T(g) define a multiplier representation of SL(2). Furthermore, computing the generalized Lie derivatives of this representation we have
(5.47)
D(2u) 0D(2v)is reducible. To decompose V(,)@ Vtv) into subspaces irreducible under the action of 4 2 ) we procede by computing the eigenvectors of J 3 which are of lowest weight, i.e., eigenvectors f of J3 such that J-f = 0 where J3 and J- are given by (5.47).T h e solutions are easily seen to be the functions h,,o(Z, w ) = N,(z
- W)S,
s = 0, 1,..., 2q,
where the N , are arbitrary constants and q J3h,,o = (S - u
-
=
min(u, v). Indeed,
J-h,,,
v)hs,0,
= 0.
I n analogy with Section 4-5,we introduce on V ( u@ ) V f va) complex symmetric bilinear form B determined by the relations B(Pk.2 ,P,,,z*)
%,*,
szpz, 9
< k,k' < 2u, 0 < I , 1' < 2v, 0
(5.48)
where 81,k = 0 if k # 1, 8k,k = 1. Thus, B(f, h) = B(h, f)E @ and + a'f ', h) = aB(f, h) + a'B(f ', h ) for all f,f ', h E V ( u@) V('), a, a' E @. T h e following lemma relating this bilinear form to the operators (5.47)is easily proved:
B(af
172
5. LIE T H E O R Y AND H Y P E R G E O M E T R I C FUNCTIONS
Lemma 5.1
(i) B(J% h )
) V(u), For all f,h E V ( u@ =
B(f, J3h),
(4 B(J+f,h ) = B(f, J-h). This bilinear form can be used to normalize the functions h8,*, i.e., the constants N , can be chosen so that B ( h , , o ,h,,J = 1. Thus,
=
N,"
'
1
j=O
(s!)2
(2u - j ) ! (2v - s + j ) ! j ! (s - j ) !
-
1.
Making use of the identity j=O
(m + k - j ) ! (n +j ) ! j ! (k - j ) !
-
m!n! (m
+ n + k + l)! + n + l)!
(5.49)
k ! (m
(Gelfand et al. [l], p. 149), we obtain Ns
=
(-1)'
[s! (2u
(224 -
+ 2v
-
2s
s)! (2v - s)! (2u
+ I)! + 2v
-
s
+ 1)1
lP
'
s = 0,1,..., 2q,
(5.50)
where the phase factor (- 1)" has been chosen for convenience. Now that the functions hs,ohave been determined, we define additional vectors hs,k by ( 2 ~ 2~ - 2s - k)! hs,' = k! (2u 2v - 2s)! (J+)'k0 , 2v - 2s, s = 0,I ,...)2q, k = 0,1)...)2u (5.51)
[ ++
+
where the operator J f is given by (5.47). Each hs,k is a polynomial of order k in z and w , and there are a total of (224+ 2v - 2s 1) = (2% 1)(2v + 1) such polynomials. Furthermore, it follows exactly as in = 0. We will show the proof of Theorem 2.3, part (iii) that J+h,,2u+2v--2S that the hs,k form a basis for V ( u ) @ V ( v ) . s
+
z!:o
+
Lemma 5.2
+
+
+
(i) J+h,,k = [(k 1)(2u 2~- 2s - k)]'/2 hs,k+l, (ii) J-h,,, = [k(2u 2v - 2s - k 1)]1/2 h s , k p l , (iii) J3hs,k= ( k - ( u v - s)) h s , k , s = 0, 1 ,..., 2q, q = min(u, v), k = 0, 1,..., 2u 2v
+
+
+
+
-
2s.
5-5. THE TENSOR PRODUCT D(2u) @ D(2v)
PROOF
+ 2v 2s - k)! = [(k + 1)(2u + 2v 2s
(9 J + L = [ k! (2u + 2v - 2s)! ]1’2 (2u
-
-
(iii)
173
(J+)k+lhs,o k)]l/zhs,k+l .
~
From the identity [J3, ( J+)k] = k( J+)k there follows (224
J3hssk
+ 2v
-
2s
-
[ k! (224 + 2v +~ 2~+ 2v2s = [ ( k 2! (2u =
-
-
-
]
J ( J )~hs,o +
]
k)! ljZ
2s)!
*
(J+)‘J3hS,,
+ 2v - 2s - k ) ! k(J+Ikhh,,, k! ( 2 + ~ 2~ - 2 ~ ) !
1’’
(224
-[
=
k)!
2s)!
(k - (U + v
-
S))hs,k.
(ii) We use induction on k. Since J-h,,, = 0 the equation holds for 0. Assume (ii) is valid for k k, where 0 k, < 2u 2v - 2s. Then
k
<
=
J-hs.k,+l
= [(ko
+ 1)(2u + 2~
from (i). T h e relation J-J+ imply
J - J + ~ =~ J, +~J ~- ~ , , ~ , ~
=
JfJ-
+
<
-
2s
-
ko)1-1’2
- 253 and
2 ~ 3 h , , ,= ~ (k,
J-J+hs.k,
the induction hypothesis
+ 1)(2u + 2v
-
2s
-
k o ~ h s . k. ,
+
Therefore, J-h,,k,+l = [(A, 1)(2u + 2v - 2s - k,)]112hs,k,. Q.E.D. Comparing Eq. (5.41) with Lemma 5.2 we see that for a fixed value of s the vectors hs,k, k = 0, 1,..., 22.4 + 2v - 2s, form a basis for the representation D(2u 2v - 2s) of 4 2 ) . Thus, the action of the operators T(g),g E SL(2), on the vectors is
+
ET(g)hs.kl(Z, 4
c
2u+2v--Zs
=
Q;;+v-S’(g)hs,L(z, 4
(5.52)
1
1=0
0, 1,..., 2q, k = 0, 1,..., 2(u + v - s), where the operators T(g) and the matrix elements Q are defined by (5.46) and (5.43). A simple induction argument utilizing Lemmas 5.1 and 5.2 yields:
s =
Lemma 5.3 s, s’ =
0 , 1,..., 2g,
B(hS,, , hs,,kr)= 6s,s’ k
=
0 , 1,..., 2(u
6k,kr
+v
9
- s),
k’ = 0, 1,... 2(u )
+v
- s’).
174
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
+
+
According to this result, the set of (2u 1)(2v 1 ) vectors {hs,k}is linearly independent; hence, it forms a basis for V ( u0 ) VtV). zmin(u,v)
C
D(2u) 0D(2v)
Theorem 5.1
@ D(2u
k=O
+ 2v
2k).
T h e Clebsch-Gordan coefficients for this decomposition, C(U,
nonzero only if 1 hS,k =
+j
=
I; .,j I r , k)
k, are defined by
C C ( u , I ; v,j
Iu
2.j
s = 0 , 1,..., 2 min(u,
+ v - s, k + s
a), k
= 0,
-
u -v)~,,~,
+a
1,..., 2(u
(5.53)
- s).
Thus,
emin(u,v) Pl*i =
1
s=o
+v
C(u, 1; v , j I -u
< I < u,
1 +j)hsJ+j+u+v-s
- s,
-v
?
< j < v.
(5.55)
T o derive a generating function for the Clebsch-Gordan coefficients, set g = exp(--by+), k = 0, in (5.52) and obtain
+
N,q(bz
1)2"-"bw
+
ztu+v-s) 1)2"-S(z
- w)S
=
1
(4)'
1=0
. [ (2u + 2v l! (2u
or
+ 2v
-y2
2s)! 2s - Z)!
-
-
+ 1)2"-"bw + - w)s = z=oc c [(2u + 2v 2S)!]l/Z . b'(-l)"(u,j u ; a,s + 1 - j v 1 u + - s, s + z - u [Z! ( 2 +~ 2v 2s l ) ! j !(2u - j ) ! (s + 1 j ) ! (2v s
N,(bz
2(u+v-s)
1)2"-S(z
-
z+s
7=0
-
-
h , , k w).
-
-
2,
-
-
-
- v)zjZus+l-j
I +j)!]'i2
*
(5.56)
At this point it is useful to note that the Clebsch-Gordan coefficients for which we have just established a generating function are exactly the
5-5. THE TENSOR PRODUCT D(2u) @ D(2v)
175
coefficients computed by Wigner for the tensor products of irreducible representations of SU(2) (the group of 2 x 2 unitary matrices with determinant + I ) (Wigner [ 3 ] , Hamermesh [l]). This is due to the fact that the unitary irreducible representations of SU(2) are obtained by restricting the representations D(2u) of SL(2) to the subgroup SU(2). We will examine this relationship in more detail in Section 5-16. T o show the connection between (5.56) and the theory of angular momentum in quantum mechanics we cast our results into a more conventional notation. Namely, we set ll = u, 1, = v, l3 = u + v - s, m, = j - u, m, = s 1 - j - v, m, = I- u - v s where 21, is a nonnegative integer and mitakes the values mi = li , li - 1,..., -li + 1, -li for i = 1, 2, 3 . Further, we introduce the 3-j coefficients defined by
+
+
c
-
(
l1 12 k, - 11 k, - 12 k, - 1, (5.58) [k,!(21, - k,)! k,! ( 2 4 - k,)! k,! (21, - k3)!]1/z * ZklWk2Xk3
O$k <2L, k,+k2+k~~1+12+13
This is a well-known generating function for the 3-j coefficients (Schwinger [I]). T h e obvious symmetry of this expression implies the existence of corresponding symmetries among the 3-j coefficients. T h e reader can derive these symmetries for himself or refer to standard references on the subject (Hamermesh ill, Wigner [ 3 ] , Bargmann [ 4 ] ) . Equations (5.54) allow us to express the matrix elements of the representation D(2u) @ D(2v) with respect to the p,,$ basis in terms of the matrix elements with respect to the hs,k basis. A standard argument (Lyubarskii [l], p. 234) yields Q:xg)Q;;,Yg) zmin (u.z.)
1
=
C(u, 1 - u;v,j- v I u
s=o *
C(u, I' - u ; v,j '
-
v
+ v -s,
I u + v - s, I'
+j'
I+j -
u
v)
-
u
-
~.l)Q~~~~;.$+,,_~(g),
-
(5.59)
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
176
where 1, 1‘ = 0, 1,..., 2u; j, j’ = 0, I ,..., 2v, and g the B functions (5.38) this relation reads
c
z m i n ( u .a)
% y ( z ) % ; ’ q z= )
s=o
C(u, 1; v , j 1 u
C(u, 1‘; v,j’ 1 u -u
where
+
E
ZI -
8 4 2 ) . I n terms of
s,
I +j)
+ v s, 1‘ +j‘)R;i;y:’+f(z), < I , I’ < u , -v < j ,j‘ < v, -
(see Section 5-14).
5-6
The Tensor Product
r,
01,
T h e local multiplier representation t, , 2u not a nonnegative integer, can be realized on the space Y2with basis h,(z) = zk, k = 0, 1, 2, ... : J3h, = ( k - U ) h , ,
J+h, = ( k
-
2 ~ ) h k + ~ , J-hk
-khk-l,
(5.60)
where J*, J3 are the differential operators (5.16). I n terms of this basis the matrix elements B$)(g),g E P, are defined by (5.22), (5.23) where the superscript u denotes the representation T, . T h e multiplier representation 7 , @ t, of P C SL(2) (2u and 2v not nonnegative integers) can be defined on the vector space Ol; 0Ol; , consisting of all functions f ( z , w),analytic in a neighborhood of the point (0, 0) E @ x @. T h e functions ~ , , ~ (w) z , = z k w i ,k, I = 0, 1, 2 ,..., form an analytic basis for Ol; @ 6Ei . T h e action of P on this space is given by the operators T(g),
where f E 6EL @ 6Ei and g E P is in a suitably small neighborhood of the identity element. I t is easy to check that the operators T(g) define a local multiplier representation of P. This action of P on Ol; 06E; induces a representation of 4 2 ) in terms of the generalized Lie derivatives
(5.62)
5-6. THE TENSOR PRODUCT
t w@ Tv
177
operating on the space V2@ V2of all polynomials in z and w. T h e Lie derivatives (5.62) are formally identical with (5.47), but they operate on different spaces and are defined for different values of u and v. As in Section 5-5 we try to decompose V2@ Vzinto subspaces irreducible under the action of 4 2 ) . Again we compute the eigenvectors f o f J 3 such that J-f = 0, where J3, J- are given by (5.62). T h e solutions are, to within a multiplicative constant, the functions s = 0 , 1, 2 ,... .
fs,o(z, w ) = (2 - w)3,
(5.63)
Thus, J3fs,0 = (s - u - ZJ) fs,o, J-fs,o = 0, for s 3 0. Let Q be the symmetric bilinear form on V2@ V2with the property
,P,,,,,)
Q(Pk.2
==
h,,, at.,z,t(kI);
where
t(k,4 = ( T h e vectors operators J*,
(-I)"%!
r(K
k k', 1, I'
=
0, 1,
(5.64)
&a**>
I! r(-2u)T( -2v) -
2u)T(l - 2v)
(5.65)
*
= 0, 1, 2 ,..., form a basis for V2@ V2 .) If the are defined by (5.62) it is easy to prove:
p k , l ,k, I J3
Lemma 5.4
(i)
Q(J3Pk,l
9
Pk,,lj) = QtPk,,
J3Pkr,1,),
(ii) Q(J+~tpr,,i P k r , ~ )= Q ( P k , l J - P k , , l * ) . T h e factor ((k, I) has been chosen just so Lemma 5.4 would hold. Q will be used as a kind of bookkeeping device to facilitate our calculations. Direct computation using (5.62) and (5.64) proves 9
s
= 0 , 1 , 2 ,....
(5.66)
I n analogy with the procedure in Section 5-5 we define vectors fs,k
=
+
T(2s - 2u - 272 K ) k, s ~ ( -22u ~- 2 4 (p)'fs,o,
+
= 0, 1,
2 ,...
fs,k
by
I
+
Eachf,,,, is a polynomial of order s k in z and w ,and there are n 1' such functions of order n in z and w . Thus, if the set { f s , k ) is linearly independent it forms a basis for V2@ V 2 . In exact analogy with Lemma 5.2 we have: .
178
5 . LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
Lemma 5.5 (k
J’fs,k
- 2U -
J%,k
= -kfs,k-i
J”fs,k
=
( k -U
-
2v
+ 2s)fs,k+i, = 0,
k, S
+S)fs,k,
V
1, 2 ,... .
A comparison of Lemma 5.5 with (5.60) shows that for fixed s the vectors k = 0, 1, 2,..., form a basis for the representation fu+v-s of sZ(2). Thus, the action of the operators T ( g ) ,g E P , on the vectorsL9,, is
fs,k,
m
[ T ( g ) f s , k ] ( z , w, =
1 B ‘ ~ ~ ” - ” ‘ ( g ) f s . Z ( z ,w),
k,
1=0
=
0, 1, 2,*-*, (5.67)
where T ( g ) and the matrix elements Blk(g)are defined by (5.61) and (5.22). Lemmas 5.4 and 5.5 can be used to evaluate the quantities Q(fs,k , fs,,k,). T h e result is:
Ns,k
( - l ) s + k ~ ! k!F ( 2 s
F(k - 2u
- 2v
-
2~
-
+
2 v ) F ( -224 - 2~ 2s - 1)F(- 2 u ) F ( - 2 ~ ) - 2v s - l ) F ( -2u s)F( -2v + s) ’
+
+ 2 s ) q -2u
+
k,k’,s,s’ = 0 , 1 , 2
,....
According to this lemma, the set {fs,k ; s, k 3 0) is linearly independent; hence, it forms a basis for VzQ Vz. Thus, ‘KzQ V. can be decomposed into a direct sum of subspaces, each subspace transforming according to an irreducible representation of 4 2 ) .
r,
Theorem 5.2
m
Q 7, c 1 0 fu+v-s s=o
We define the Clebsch-Gordan coefficients E(u, I ; v , j j s, k) nonzero k = I j , by only if s
+
+
s+k fs,k =
2 E(u, I; v,
1=0
f
-
1 s, k)pZ.s+k--l
k,
7
2 O.
(5.68)
Thus, Q ( f s , k , pz,d
=
E(u, I ; v,j I s, k ) t ( l , j )
%,s+k-Z
;
This relation allows us to invert (5.68) and obtain
s,
k , l , j 2 0.
(5.69)
tu 01,
5-6. THE TENSOR PRODUCT
179
A generating function for the vectors fs,k and, thus, for the ClebschGordan coefficients E(.) follows easily from (5.67). Evaluating this equation for the case where k = 0 and g = exp(-by+), we have
or
*
+K - j
E(u,j; V , s
1 S, k)Z'Wsfk--j,
(5.71)
where 1 bx 1 < 1, 1 bw 1 < 1, s >, 0, 224, 2v not nonnegative integers. Comparison of this expression with the generating functim (5.21) for the matrix elements B,,(g) yields the interesting relation
+
(-l)S+l(l
Q2,--s
1 + 1; --b) + 1) 2s ) bLE(u,I ; u , s + k - 11 s, k ) ,
s! F(-Z, 7
-2u
+ s; s
r(s ~
=f( + 2u
k=O
2v k
-
-
I
Ibl
<
< 1.
(5.72)
If s - I + 1 0, the left-hand side of (5.72) is defined by (A.5). Equations (5.69) allow us to express matrix elements of t, 0 7, with respect to the p,,? basis in terms of the matrix elements with respect to the fs,k basis. T h e result is
*
E(u, 1'; v,k' lj, I'
+ k'
-
j)B~~~~~,j~,+k,-~(g),
g E P,
or
I, l', k , k' 2 0;
180
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
T h e tensor products 1, 0 I,, 24224) 0 f, , and D(2u) @ 4, can also be studied by the above methods but this will be omitted since the results are similar to those we have already obtained.
5-7
Differential Relations for the M a t r i x Elements
T h e matrix elements of the representations D(u, mo), t,, J,, and D(2u) have been shown to be analytic functions of the group coordinates (a, b, c ) in a neighborhood of the identity element e of SL(2). Thus, if 9 is the complex vector space consisting of all functions of the coordinates of SL(2) analytic in some neighborhood of e, it follows that the matrix elements A j k ( g ) , B,,(g), C j k ( g ) , Dik(g), determined in Sections 5-1 to 5-4, are members of 9.There is a natural action of SL(2) on 9as a local transformation group. If g’ E SL(2)let P ( g ‘ ) be the linear operator f r o m 9 to 9defined by [ P ( g ’ )f ] ( g ) = f ( g g ’ ) for allf 6 9and g E SL(2) such that g’ and gg’ are in the domain off. From this definition there follows the relation [P(g,g,)fl(g) = [P(g,)(P(g2)f)I(g)
for g, , g , ,g in a sufficiently small neighborhood of e. T h e action of P on the matrix elements is given by
c Czk(g’)cdg)j
(5.74)
m
[P(g’)C,,l(g)
=
C,,(gg’)
=
1=0
j , k nonnegative integers;
c Dlk(g’)D,l(g), 2u
[P(g‘)Q,l(g)
=
%C(gg’)=
1=0
2u a nonnegative integer, j , k = 0, 1,...,2u;
defined for g’,g in a sufficiently small neighborhood of e . Thus, for fixed j , the functions { A l k }form a basis for the representation D(u, mo),
5-7. DIFFERENTIAL RELATIONS FOR MATRIX ELEMENTS
181
the functions {Bjk}form a basis for t, , the functions {Cjk} form a basis for I,, and the functions {Dj,} form a basis for D(2u). T h e Lie derivatives J+, J-, J 3 of the multiplier representation P are defined by d J+f(g) = ~ [ W ~ t%+)fl(g) X P
I
i=O
for every f E 5and every g in the domain off. It is a consequence of the discussion of Section 2-2 that these Lie derivatives satisfy the commutation relations
[J3, J*I
=
iJ',
[J+, J-I
=
2J3
and that they act on the matrix elements as follows:
J3Ajk(g) = (mo + k)Ajk(g) (5.76)
(5.78)
182
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
T h e above expressions yield recursion relations and differential equations for the matrix elements, which can be evaluated by computing the Lie derivatives J*, J3 from (5.75). Instead of the coordinates (a, b, c ) for a neighborhood of e in SL(2), however, we shall find it more convenient to adopt the coordinates [a, 8, w ] such that a = e(W+B)P
W cash 2 '
b
= e(W-B)/2
W sinh , 2
cash w = 2bc
W sinh 2
c = e(-W+B)P
+ 1.
(5.80)
T h e complex coordinates [a, 6, w ] are not defined for group elements such that bc = 0, so they cannot be extended over the entire group manifold. Furthermore, as we have shown in Section 5- 1, the coordinate tranfformation (5.80) is not one-to-one since distinct points in the [ a 8, w ] coordinates correspond to the same point in the ( a , b, c) coordinates. For group elements such that bc # 0, however, the Jacobian of this coordinate transformation is nonzero and the transformation is locally one-to-one and analytic. T h u s if g E SL(2) has coordinates [a, 8, w ] and
is in a sufficiently small neighborhood of e it is easy to verify that gg' has unique coordinates [a1 , w l ] where
Pl, cosh w1 = cosh w (a'd' + b'c' + eaa'b' tanh w + e-add' tanh w), (eW(cosh w + 1) + c' sinh w)(e-W(cosh w 1) + b' sinh w) (d'(cosh w + 1) + eab' sinh w)(a'(cosh w 1) + e-Bc' sinh w) (a'(cosh w + 1) + e-Bc' sinh w)(a'(cosh w 1) + e - k ' sinh W ) I ~ / ~ eai = ea [ (d'(cosh w + 1) + eab' sinh w)(d'(cosh w 1) + b'ea sinh w) -
P I = , . [
-
-
-
(5.81)
From this result and the definition (5.75) of the Lie derivatives we obtain
C1,,=(22-
a2
a
a22
az
1)-+2z-+---
1 22 -
1
a2
a2
am2
8/32
+ 22-)
~ - - -
a2
am ag
5-8. TYPE B REALIZATIONS OF D(u, m,)
183
where z = cosh w. As in Section 5-1 we assume that x takes values in the cut complex plane where the cut runs along the real axis from - co to 1. I n particular, if x > 1 then (9- I ) l l 2 > 0. According to the relations (5.12), (5.24), (5.31), (5.38), the matrix elements A,(g), B j k ( g ) ,C,,(g), D,(g) can be expressed in terms of the functions 23>m(z). Substituting (5.12) and (5.82) into Eqs. (5.76) and simplifying we obtain the relations
+
+ u + l)(m - ~ ) B ; ~ + l ( z ) ,
(m
a
( ( 2- 1) - + 22 - ax2 az a2
= u(u
+ l)B>"(z),
2zmr + r 2 22
+ m2
-1
) %3:."(4
for all complex numbers u, r, m such that u f m,u f r, are not integers and r + m is an integer. Similarly, from Eqs. (5.77) we find that (5.83) remains valid for all r, u - m are nonnegative complex numbers r, m,u such that u integers and 2u is not a nonnegative integer. Finally, from (5.79) we again obtain equations (5.83), valid now for all nonnegative integers m are integers and 2u and all numbers r, m such that u - r , u 0 u - r , u + m, 2u. T h e derivation of recursion relations from (5.78) is left to the reader.
+
<
5-8
<
+
Type B Realizations of D(u, m,)
I n this section we study realizations of D(u, m,) such that J*, the type B differential operators
J3
are
derived in Section 2-7. I t will be more convenient, however, to introduce new variables T , x defined by T = y - iz-12, x = -ieciX. T h e type B operators then become (5.85)
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
184
<
T o find a realization of D(u, m,), (0 Re m, < 1 , u f m, not integers) in terms of the operators (5.85) we look for nonzero functions fm(z, T) = Znl(z)emTsuch that J3fm =
mfm
,
E
(m - u)fm+i ,
J-fm
=
-(m
+ uI.fm-1
(5.86) + J3J3 - = ~ ( +u 1 ) f m S where 5' = {m,+ n: n an integer}. I n terms of the functions Ci,ofm
for all m
J+fm =
=
(J'-
J3)fm
Z,(z) these relations become
(5.87)
As shown in Section 2-7, Eq. (iii) has as solutions the functions Zm(z)= (2qz)u+le-Q2 ,F,(u
-
ZA(z) = ( 2 g ~ ) - ~ ,F,( e - ~-u~
m
-
+ 1; 2u + 2 ; 2qz),
m; -2u; 2gz).
Expressed in terms of the generalized Laguerre functions, there are solutions
(5.88) From the form of these solutions it is clear that without loss of generality we can set q = 4. Second, focusing attention on solution (I), we note that it is a generalized Laguerre function multiplied by the factor (z)u+1e--z/2.It would be more convenient for the study of Laguerre functions if we could find Lie derivatives whose eigenfunctions were of the form fk(z,T) = Lz:::\(z)emT, (5.89) without the extraneous factor. By transforming the type B operators we shall bring about this situation. the space of all functions analytic in a neighborhood of Denote by the point (zo,TO) = (1, 0). ( T h e exact choice of (zo,TO) is not important.) : ( az )simultaneous emT eigenT h e function f,,,(z,T) = z u + l e - - z / ~ ~ ~ ~ ~ ~ is
5-8. TYPE B REALIZATIONS OF D(u, m,,)
185
function of the operators J3, C‘l,o and is an element of 9. Define the , = z”f1e-z/2f(z,T) for all linear mapping cp of F onto 9 by [ q f ] ( z T) f E ,F. T h e mapping can be considered as multiplication by the function ~ ( z=) zU+1e-z/2.Similarly, the map 9-l: F -+ F, the inverse of cp, is defined by [cp-lf](z,T) = z-’-lez/zf(z, T ) . Under cp-1 the eigenfunction f,, is mapped into fk where f L ( z , T) = [(p-lfm](z,7) = LgE:Li(z)emT is the function (5.89). As was shown in Section 4-8, Lemma 4.4, 9 induces a Lie algebra isomorphism J - J” of the linear differential operators on 9such that (J”)(cp-lf) = c p - I ( J f ) for all f E F.T h e operators (J+)”, (J-)Q (J3)” can be computed from Eqs. (4.82) and (5.85):
(5.90) (J3)Q
=
a
tat
where t = e7. T h e operators (5.90) automatically satisfy the commutation relations for the generators of 4 2 ) so they can be used to construct a Thus, we look realization of the representation D(u, m,) on 9. for nonzero functions fm(z,t ) = Zm(x)tm in 9,defined for all m E S = {m, n ; n an integer}, such that Eqs. (5.86) are satisfied, where now the differential operators are given by (5.90) and we have dropped the superscript cp. I n terms of the functions Z,(z) these relations take the form
+
d
(z -z dz
+ ( m + + 1)) Zm(z)
d ( z - - (m dz
(2
~
u
-
=
(m
1)) Zm(X) = -(m
d2 dz2 + (2(u + l)-z)
d
dz
+ (m
-
+
u -
U)Zm+1(4,
~
U ) Zm-1
(z ) ,
(5.91)
1
I) P ( z ) = 0
for all m E S. T h e solutions are generalized Laguerre functions. I n particular, Z,(z) = LE:;:;(z) (5.92)‘ satisfies (5.91) for all m E S. None of these solutions are polynomials in x since m - u is not an integer.
186
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
T h e above remarks show that the vectors fm(z,t ) = LgY$i\(z) tm form a basis for a realization of D(u, m,). This realization can be extended to a local multiplier representation T of SL(2) on 2F. Applying Theorem 1.10 to the generalized Lie derivatives (5.90) we can determine the action of the operators T(g),g E SL(2), as follows: [T(exp -b$+)f](z, t ) = ( 1
t + bt)-u-lebztl(l+bt)f (6% i&>
valid for all f E 9, (2,t ) in the domain off, and sufficiently small values Oflbl,lCl,I~l. If g E SL(2),Eq. (1.15), with d # 0 it is easy to verify the relation g
=
(exp -b'$+)(exp
-c'$-)(exp
where b' = b/d, c' = cd, eT1l2 = d-I , 0 operator T(g) is given by
zt
at
< Im
+c
T'$~)
T'
< 477.
Hence, the
I clat I < 1 , 1 bt/d I < 1,
(5.93)
for f E 2F and g in a small enough neighborhood of e so that the above expression is uniquely defined. T h e matrix elerrients of this multiplier representation with respect to the analytic basis {fm) are the functions A,,(g), Eqs. (5.10), (5.12). Therefore, we have the identity
5-9. TYPE 6 REALIZATIONS OF
187
fu
which simplifies to
+
This relation is valid for all p, v E Csuch that Y and p v are not integers. 1 \< 0 the hypergeometric function F is defined by the limit When I (A.5). I n general the right-hand side of (5.94) converges whenever the left-hand side does. We shall investigate two special cases of this identity. If a = d = t = 1 and c = 0, then
+
161
If
a =
d
=
t
=
1 and b
=
0 there follows
Each of these expressions is valid for all p, v not integers.
5-9
< 1.
Type B Realizations of
E
@such that v and p
+ v are
t,
r,
T o find a realization of the representation of s1(2), 2u not a nonnegative integer, by type B operators it is most convenient to transform these operators into a new set of Lie derivatives especially adapted to solution (2) of (5.88) ( q = 4). Thus, we set g)(z)=,z-ue-z/2 and compute the operators (J*)'p, (J3))" where J*, J3 are given by (5.85). (For this choice
188
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
of ~ ( z the ) , common eigenfunctions of (C,,$' and ( J3), will be constant ) ( zm)E S.)T h e result is, Eq. (4.82), multiples of L ~ ~ ~ - ' tm,
= t-1
(J-)Q
a
(z -az
(J3)Q
=
-
t-
a
at a
-
(5.95)
u l y
t -.
at Dropping the superscript q , we see that to construct a realization of t, with the operators (5.95) it is necessary to find nonzero functions fwL(z, t ) = Z,(z) tm, m E S = {-u n, n >, 0}, with the properties
+
J3fm =
mfm
J+fm
=
Ci,nfm = (J+J;
( m - u)fm+i,
+ J3J3
- J3))fm
J-fm
1
= U(U
-(m
+
+ u)fm-1>
(5.96)
1)fm
for all m E S. These conditions will be satisfied if and only if the functions Z,(z), m E S , satisfy the relations d (2 - - z ( m - 4)Z,(z) = ( m - u)Zm+l(z), dz
+
+ 4)Zm(z) -(m + U)Z d2 d (.p -(2u + +(m + ). l G ( z ) dz d
(z- (m dz
=
2)
m-1
(z ) ,
=
(5.97)
0,
where Z-,-,(z) z 0 on the right-hand side of these expressions. Equations (5.97) determine the functions 2, to within an arbitrary constant. Indeed, for m = -u, z d / d z Z - , ( x ) = 0, which has the solution Z-,(z) = c, c a constant. We normalize this solution by setting c = 1. For m = -u n, n 3 0, the functions Zm(z)can be defined by the relation
+
A straightforward induction argument using (A.13) yields
where the functions L,!-2u-1)(z)are generalized Laguerre polynomials. By definition the functions
5-9. TYPE B REALIZATIONS
OF 1%
189
satisfy the relations J”fm
=
mfm ,
J+fm
=
(.z
-
u)fm+l
,
J-fL
= 0.
Moreover, an induction argument utilizing the commutation relation [Jf, J-] = 2J3 shows that the functions f m actually satisfy all of the relations (5.96). Th u s the solutions (5.98) satisfy Eqs. (5.97) for all m = -u + n E S , and we have constructed a basis for a realization of the representation 7 , of 4 2 ) on 3. This realization of 7 , can be extended to a local multiplier represenAs is easily verified, the operator T’(g),g E SL(2 tation T‘of SL(2) on 9. is obtained from (5.93) by replacing u with -u-I. Thus,
+ bt)”(a + c/t)”ebzt/(d+bt) zt at + c * f((at + c)(d + 6 t ) ’ -1’
[T(g)f](z, t ) = ( d
I c/at 1 < 1, I bt/d 1 < 1,
(5.99)
for f~ F and g in a small enough neighborhood of e such that the right-hand side of this expression is uniquely defined. T h e matrix elements of the multiplier representation T‘ with respect 0, are the functions B,,(g), I, K 3 0, to the analytic basis {f-,+,}, n (5.22)-(5.23). Therefore,
>
00
[T’(g)f-,+,l(z,t )
C Bzk(g)f--u+i(z,t ) ,
k = 0 , 1,2,*-..
1=0
This leads to the identity
1 bt/d < 1, ~
>
d
=
(1
+ bc)/a,
(5.100)’
valid for all integers k 0 and for all u E @ s u c h that 2u is not a nonnegative integer. When I - k + 1 < 0 we use the limit (A.5) to define the hypergeometric polynomial on the right-hand side of this identity.
190
If
5. LIE THEORY A N D HYPERGEOMETRIC FUNCTIONS a =
d
=
t
=
1, c
=
0, the above identity simplifies to
Ibl
< 1.
For k = 0 this last expression becomes a well-known generating function for the generalized Laguerre polynomials: (1 -b)Zue-bz/(l-b)
C blLI-zu-l)(z), m
=
i=n
I b 1 < 1.
(5.101)
= 1.) (We have replaced b by --b and used the fact that Lb-zu-l)(~) A simple consequence of (5.101) is L(-2U-1'(0)
r(z- 2 4
z! 1 7 -2 4
~
1
When a
=
d
=
t
=
1, b
=
z = 0 , 1 , 2 ,....
0, (5.100) becomes
Consideration of type B operator realizations of the representation 1, would lead to the identity (5.100) again and give no new results. Hence, we shall not embark on this study. Finally, as the reader can verify for himself, there are no type B operator realizations of the finite-dimensional representations D(2u), 2u a nonnegative integer, of sl(2). To construct a realization of D(2u) operating on a space of functions of two complex variables it is necessary to use type A operators.
5-10 Weisner's Method for Type 0 Operators
So far the modified type B operators (5.95) have been used to construct identities for special functions which are simultaneous eigenfunctions of J3 and C l , n . However, these operators can also be used to derive identities for eigenfunctions of Cl,o which are not eigenfunctions of J3. We make the following observations. Iff(., t ) is a solution of the equation C,,,f = u(u I ) f , i.e.,
+
(5.102)
5-10. WEISNER’S METHOD FOR TYPE 5 OPERATORS
then the function T‘(g)f , g E SL(2), formally defined by z [ T ‘ ( g ) f ] ( zt, ) = (d bt)u(a +
+
at
191
+c
+
is a solution of the equation C,,,[T’(g)f] = u(u l)[T’(g)f]. This remark is true whenever the formal expression for T‘(g)f can be interpreted as an analytic function in (2,t ) , and is a consequence of the fact that C,,? commutes with J+, J-, and J3. Furthermore, iff is a solution of the differential equation (xiJ+
+ xzJ- + x 3 J 3 ) f ( z ,t ) = x f ( z ,t )
for complex constants x1 , x2,x3,A, then Eq. (1.25), Section 1-2, shows that T’(g) f is a solution of the equation ((abx, - b2z2
+ a2xxl)J++ (-cdx,
- C’X,
+ d2x2)J-
+ ((ad + bc)x3 - 2bdx2 + 2acx,)J3})IT’(g)f]
= X [ T ’ ( g ) f ] , (5.103)
+
for g E SL(2),Eq. (1.15). Finally, if Cl,of ( z , t ) = u(u 1) f ( z , t ) and f has a convergent expansion of the formf(x, t ) = C h,(z) tm, then the
expansion coefficients are solutions of the Laguerre differential equations
These observations provide powerful tools for the derivation of identities satisfied by Laguerre functions. (We are now concerned with the operators (5.95) but the above comments apply as well to the other two variants of type B operators studied in this section.) As an application of these remarks we choose f to be a solution of the simultaneous equations
T h e general solution of (ii) isf(z, t ) = t-ueth(zt) where h is an arbitrary function of xt. Substituting this solution into (i) we find that h must satisfy the equation d2h(w) dh(w) w --2u _ _ +h(w) = 0. dw2 dw
192
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
This is a modification of Bessel’s equation and has a solution, h(w) = w(2u+1)/2J-2u--1(2w1/2), which is an entire analytic function of w. Thus,f(z, t ) = t-Uet(zt)(2U+1)/zJ_2U-1(2(zt>112) is a solution of (i) and (ii). According to the above remarks, the function [ T ’ ( g ) f ] ( z ,t ) = t-”(d
t c] (,t)‘2u+l’/2 + bt)-l exp [( a +d +bz)t bt
satisfies the equations C,,o[T’(g)fl (-b2J+
$- d 2 J -
=4 u
+ l)tT‘(g)fIY
2 b d J 3 ) [ T ’ ( g ) f ]= - [ T ’ ( g ) f ] .
~
Since w-mJm(w) is an entire function of w for all m has an expansion in t of the form
E
(5.104)
@, [ T ’ ( g ) f ] ( z ,t )
m
Thus, T ’ ( g ) f is a generating function for the coefficients h,(g, z). We will determine these coefficients. Substituting the expansion into the first equation (5.104) we find that h,(g, z ) is a-solution of the Laguerre differential equation
T h e function [ T ’ ( g ) f ] ( z t, ) is regular at z
=
0. I n fact,
Thus, the functions h,(g, z ) must be multiples of generalized Laguerre polynomials: I = 0 , 1 , 2 ).... h,(g, z ) =j,(g)L;ZU-’(z), We could now use the second equation (5.104) to derive recursion relations for the j , ( g ) . However it is simpler to proceed in a different manner. So far we have obtained the relation (d
1bt)-l
exp
F( a +d +bz)tb t
c
2u-1
Q
=
Cj,(g)L;2”-1(z)t‘,
z=o
d
=
(1
+ bc)la,
d I bt
1 bt/d I < 1.
193
5-10. WEISNER'S M E T H O D FOR TYPE 6 O P E R A T O R S
At z
=
0 this relation becomes
T h e above expression is reminiscent of the generating function (5.100) for generalized Laguerre polynomials. I n fact when k = 0, (5.100) becomes
Comparing these two expressions we find 1
+ bc
b(l
+" bc)I '
1 = 0 , 1 , 2 )...,
where 2u is not a nonnegative integer. For c computation is the identity
m
=
0, the result of this
71
I abt I < 1. If a
=
1, b
while if a
=
=
(5.105)
0, (5.105) becomes
iy1/2,b
=
iy-1I2, it reduces to the Hille-Hardy formula
As a final example we apply T'(g) to the functionf(z, t ) = L;Y;'(z)tm, m, u E @. Clearly C,,J = u(u l)f, J"f = mf. I n the case where g is
+
194
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
in a small neighborhood of e we have already obtained identities for f from the representation theory of sI(2), Eq. (5.94). However, the choice 0 g=i1
-1 1)
violates the conditions of (5.94). I n this case we can write T'(g)f in the form t-y1
-
which has an expansion in terms of t-u+l, I = 0, 1,2,... . T h e coefficient of t-%+l is c , L ; * ~ - l ( x )and the constants cl can be determined from the expansion by setting x = 0. T h e result is
It1
a generating function for the generalized Laguerre polynomials. As the reader can prove using (5.103), the examples studied here are inclusive in the following sense: Lemma 5.7 (Weisner) If f is a solution of the equation (xlJ++x, J-+x, J3) f ( x , t ) = h f ( z , t ) for complex constants xl , x2 , x, , h such that x2 4x,x, # 0 then there exist p1 E @, g , E SL(2), and a function h,(x, t ) such that f = T'(gl)h, and p1J3hl(z,t ) = hhl(z, t). Moreover, if x," 4x1x, = 0 there exist p, E Q, g , E SL(2)and a function h,(z, t ) such that f = T'(g,) h, and p2J-h2(z,t ) = hh,(x, t). Lemma 5.7 is also applicable to the work of Section 5-15.
+
+
5-11
Type A Realizations of D(u, m,)
T h e type A differential operators determined in Section 2-7 form a basis for an algebra of generalized Lie derivatives isomorphic to sI(2), which operates on a space of functions of the two complex variables x, y:
5-11. TYPE A REALIZATIONS OF D(o, m,)
195
In terms of the new variables t , z and a new constant r defined by t = ieu, z = cos x, r = -is, the type A operators take the form
T o construct a realization of therepresentation D(u, m,) (0 < Rem, < 1 , & m, not integers) using the operators (5.106) we must find basis vectors fm(z,t) = Z,(z) tm such that
u
JYm
=
mfm
9
Ci,ofm
(m - u)fm+i
J+fm
=
=
(J+J-
1
+ J3J3 - J3)f
-(m
J-fm m - u(u
+
+
~)fm-r9
(5.107)
1)fm
+
for all m E S = (m, n: n an integer). These conditions are fulfilled if and only if the Z,(z) satisfy the relations
for all m
E
S. As shown by Eqs. (5.83), the functions Zm(z)= r ( u
+ m + 1)82"(z),
m
E
S,
-+
satisfy these relations for m r an integer. However, it is easy to verify r is not an directly that the above functions satisfy (5.108) even if m integer. (As usual we assume these functions are defined in the z-plane cut along the real axis from -co to + l . I n (5.108) the branch of the square root is chosen so that (z2 - 1)1/2 > 0 if z > 1 . ) Thus, for a m + 1) 8;m(z) tm fixed complex number r the vectorsf,(z, t ) = r(u form a basis for a realization of the representation D(u, m,) of $42). W6 can use this realization by type A operators to induce a local multiplier Here 9is the space of all functions of representation T of SL(2)on 9. the complex variables z, t , analytic in a neighborhpod of the point (say) (z", t o ) = (2, 1). (Th e exact choice of (Z", to) is not important.)
+
+
196
5 . LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
According to Theorem 1.10 and (5.106) the operators T(g) on 9are determined by z2 - l ) l I 2 -
[ ~ ( e +3)fi(z, x ~
t)
=f
1)1/2
+ bt ( z + 1) + bt(z - 1)
IT’*
w,
(z,
valid for f E 9, (2,t ) in the domain off, and j b small. If g E SL(2), d f 0, then g = (exp -b’f+)(exp
for b‘ = b/d, c’ defined by
= cd,
-c’$-)(exp
< Im
eT’/2= d-l, 0
T’
1 , I c 1 , I T I sufficiently
T ’ f 3 )
< 477.
Thus, T(g) is
[T(g)fl(z, 1) =
[T(exp -b’f+)T(exp
-
(d(z2
~
-c’f-)T(exp
T ’ f 3 ) f ] ( Z , t)
l ) l P 4-ct-yz - l))(a(z2 - l)l/Z 1)1/’ bt(z 4 l))(d(z2 - l)lI2
+
+ ct-l(z + 1)) + bt(z 1)) -
With respect to the analytic basis {fin(z,t ) } the matrix elements of this local representation are the functions A,&) given by Eqs. (5.10), (5.12). Thus,
c m
[T(g)fm,+rl(~* t)
=
&k)fmo+Lh
t),
0, i l , * 2
,... .
E = -C C
k
=
(5.1 10)
5-11. TYPE A REALIZATIONS OF D(u, m,,)
197
Rather than attempt to work out the detailed consequences of (5.110) we consider only a few special cases. From Eqs. (A.9), (iii) it is clear that the function Bz~"(z)is analytic and single valued in the right-half plane Re z > 1. (If r rn = 0, B:m(z) is analytic and single valued for Re z > 0.) For simplicity we shall restrict ourselves to the half plane R e x > 1. If a = d = t = 1 , c = 0, Eq. (5.110) can be written in the form
+
valid for Re z we have
> 1, I b2(z+ l)j(z - 1)1 < 1. If a
=d =t =
1, b = 0,
+ l)/(z
for Re z > 1, 1 cz(z b = c = sinh w/2, p
=
- 1)1 < 1. Finally, for a = d = cosh w/2, cosh w ,t = e i w , there follows the identity
+
+
l)(p l>-'(z - l)-' 1 < I,, where Re z > 1, Re p > 1 , 1(p - l ) ( z a: real. If rn = r = 0 this last formula reduces to the well-known addition theorem m
198
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
for generalized spherical harmonics, valid for Re x > 0, R e p > 0, 1 arg(z - 1)1 < g, 1 arg(p - l)] < z. Here, bt(z) = B$-'(x), B,(x) = !B;,O(x). I n all of the above equations u f m are not integers. Another interesting variant of the identities (5.110) is obtained by 1. We define functions restricting z = x to the real interval - 1 \< x Pi*"(x) on this interval by
<
p',"(,) = e-Z"" U
m+r) / 2 1 9 3 T , r n (x
= w I+( 1 +x
(rn-?.)/'Z
+ i . 0)
1-
(rn+V)/2
+
where !B>"(x i 0) is the limit value of b>"l(x) as z approaches x on the upper side of the cut between - 1 and + I . By inspection these functions have the properties P y y x ) = P,m,'(x),
= PT';l1(x).
P>,(X)
Moreover, PE.-"(x) = P;(x), PE*o(x)= Pu(x), where Pc(x), P,(x) are Legendre functions (ErdClyi et al. [I], Vol. I). T h e reader can derive identities for the functions Pi."(x) directly from (5.110). U p to now we have used type A operators to obtain identities for the B functions. However, these operators can also be employed to derive identities directly for the hypergeometric functions. T h e simplest way to proceed is to return to the original form of type A operators established in Section 2-7:
22 cl,o= - 2x2
T h e function h,(z,y)
=
z =
(1
-
z)(m-Q)/2z-UF(m - u , -q
sec2(x/2),
=
mh, ,
u;
-2u; %)emu;
1 > 1 z 1 > 0,
is a simultaneous eigenfunction of J3h,
-
J3
and Cl,o :
C,,,L
=
u(u
+ 1)hm .
We will transform these type A operators into new operators with simultaneous eigenfunctions of the form hk(z, t ) = F(m--u,-q--u; - 2u; z ) tm.
5-11. TYPE A REALIZATIONS OF D(u, m,)
199
T h e procedure for carrying out this transformation is clear. We introduce a new variable t defined by t = ( 1 - z)ll2eV and the function ~ ( x= ) x-u(1 - x)-q/z. Then [cp-lh,](z, y )
= F ( m - u,
-4
-
u;
-224; z)t”
= hA(z, t).
T h e transformed operators (J3)’, (J+)”, (J-)’ in the variables x, t are defined by Eq. (4.82) and are easily computed to be
By construction these operators satisfy the usual commutation relations for the generators of $42) so they can be used to form a realization of the representation D(u, m,). T o construct such a realization we must determine nonzero functions fm(z,t ) = Zm(x)tm, m E S = {m, k: k an integer}, such that
+
+ m u1 z,(z) ( m u)z,+,(z) ( 4 1 - 4 -&+ + u ) ( m + 4)G d z ) -(m + d
=
-
(z d
=
~
d2
(z(1 - z ) 2 dz
+ [-2u
-
-
d z(m - q - 2u +1)] -(u dz
~
m)(q
U ) Zm-1
(z )
+ u ) ) Zm(z)= 0,
for all m E S. These equations have the following solution, regular at z = 0, (A.7): Zm(z) = F(m - u, -u - q; -2u; z). Thus, the functions fm(x, t ) = F(m - u, --u - q ; -224; x) tm, m E S , form a basis for a realization of the representation D(u, m,) by the generalized Lie derivatives (5.1 12). Clearly these operators induce a local multiplier representation T’ of SL(2) on the space GZ of functionsf(z, t ) analytic for (x,t ) in some neighborhood of (zo,to) = (0, I). T h e operators T’(g),g E SL(2),Eq. (1.15), acting on GZ can be computed from the results of Theorem 1.10: . c(z - 1 ) n+’ [T’(g)fl(z,t ) = (d bt)” ( a (a t
+ G)-q -)
+
-f
zt
[(d
+ bt)(at
I bt/d I < 1, I C / U ~1 < 1, I C ( Z
-
I)/at j
1)) ’
+ at
ml’
c
- c(z -
< 1, d = (1 + ~ c ) / u . (5.113)
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
200
With respect to the analytic basisf,,(z, t ) computed above, the operators
T'(g) have matrix elements A&), Eq. (5.10). Thus, m
CT'(g)fm,+k1(z,t )
=
1
ALkk)frn,+l(%
I=--m
t),
which reduces to the identity
*
(1 m
=
+ Tply)-"(l + T)mfp-u(lr ( - v + m + 1) F ( v
( Z - I)T)-&
TpL
l=-m
r(-v
im
+ 2, p; v ; z ) ,
+ -I + 1)
-
m
+ 1;y)
I , m; --I
q--I + 1) -
*F(m j T-'y
1 < 1, 1 T I < 1, 1 (Z - 1). 1 < 1,
2
# 1, 1 T-'y
-
z
1 < 1, (5.114)
+
valid for all m, p, Y, E @ such that m, -v m, are not integers and -Y is not a nonnegative integer. (The last restriction on v can be avoided if we divide both sides of the identity by I'(v).) If y = 0 this identity becomes
If, on the other hand, we set y = XT and let F (m, p ; v ;
5-12
1- z
(1 Ix ) - ~=
T
-+0, we obtain
f (-# (" +;1) F(m + I , p; -
v; z),
1=0
Type A Realizations of T,
T o construct realizations of the representation I,,, 2u not a nonnegative integer, using the type A operators (5.106) we must find nonzero functionsf,,(z, t ) = Z,,,(z)tm, m = -u, - u 1 ,..., such that
+
JYm
J+fA = ( m - u ) f m + l 1 J%, = -(m + u)fm-1 Ci,ofm = (J+Jp IJ3J3 - J3))fm = U(U t 1)fm
mfm
1
2
*
5-12. TYPE A REALIZATIONS OF tu
201
T h e above conditions are equivalent to Eqs. (5.108) for the ZJz) where, on the left-hand sides of these equations, m takes the values -u, -u 1, --u 2, ... . From (5.83) it follows that these relations are satisfied by
+
+
Ll(4 = r(-l>’+m ( m -u)
B-r,-m( z ) ,
m = -u,
-ti
+ 1,...,
(5.115)
for all r E fi? such ‘ that u - r is a nonnegative integer. Moreover, the functions (5.115) actually satisfy the required relations for all r E @, as can be verified by explicit computation. Thus the vectors
form a basis for a realization of the representation 7 , by the type A operators (5.106). These Lie derivatives determine a local multiplier representation T of SL(2) on 9 by operators T(g),g E SL(2),Eq. (5.109). With respect to , Fz 3 0}, the matrix elements of this local the analytic basis ( f - - u + k ( ~t), representation are the functions Blk(g),Eqs. (5.22)-(5.24). Hence,
Since %:m(x) is analytic and single valued for R e x identities are easily obtained from (5.1 16). If a = d = t = 1, c = 0, then
+
valid for Re z > 1, I b*(x I),(x - l)l < 1 and u integer. If a = d = t = 1, b = 0, there follows
Rez
> 1 the
+m
following
a nonnegative
> 1 , I c2(z + l)/(z - 1 ) j < 1,
202
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
where u + m is a nonnegative integer. For a = d 6 = c = sinh(w/2), p = cosh w , t = e i u , there follows
+
=
cosh(w/2),
+
valid for Re z > 1, R e p > 1, I(p - l)(x l)(p I)-'(z - l)-I 1 < 1, CY. real, and u m a nonnegative integer. There is a special case of the identity (5.116) which is worthy of consideration in its own right. If r = 0, u = z(z2 - 1)-1/2 the basis functions Bt,"-k, considered as functions of v, are multiples of (1 - u2)-"/2C~"(v) where the C;" are Gegenbauer polynomials, (A.9). Rather than obtain identities for Gegenbauer polynomials by manipulating (5.116) we will start at the beginning and derive the identities from a realization of the representation t, of 4 2 ) . T h u s we introduce the new variable u = x(z2 - 1)-lj2 into expressions (5.106) for the type A operators. I n terms of the variables ( u , t ) the differential operators become
+
J3
=
a
t -at ,
J*
=
t*l ((v'
-
a
1) av j
u t
-),ata
(5.117)
where we have set r= 0. T o realize the representation t,, 2u # 0, 1,2 ,..., with these operators, we must construct basis vectors f--u+k(u,t ) = W-,+,(v) t - " i k , k 3 0, such that
T h e functions W-,+,(v) are determined to within a nonzero multiplicative constant by these conditions. Indeed, J-f-, = 0 implies (v2 -
d W-, 1) -- U V W - , dv
= 0.
5-12. TYPE A REALIZATIONS OF
tu
203
This equation has the solution W-,(v) = (v2 - 1)--u/2,unique to within k 3 0, can now be a multiplicative constant. T h e functions W--u+k(v), defined recursively by the formula W-u+k(~)t-u+k = (-2u
+ K - 1)-l J+(W-u+k-l(~)t-u+k-l)
From the explicit expression for the differential operator J+ it is easy to verify that the functions K u t k ( ' u ) can be written in the form
where C," is a polynomial in ZI of order k. Furthermore, it follows from the commutation relations of the operators J*, J3 that the vectors f--u+k('u,t ) = W - u + k (t-u+k, ~) k 0, defined by (5.119) do in fact satisfy all of the conditions (5.118) and form a basis for a realization of t, . Writing conditions (5.118) in terms of the polynomials C,"('u) we find
>
[(I
d
- v2)
-u(k
d [(I - v2)dv
[(I
- v2)
d2
+ ak] C,.(v)
- -(1
dv2
1
2u) C,"(v)
~
-
.
=
d 2u)v dv
=
-(k
( K - 2u
+ k(K
-
+ l)c;;l(v),
-
l)c;:l(v),
2 4 1 C,.(v)
= 0.
This realization of t, can be extended to a local multiplier representation T of SL(2). T h e action of the operator T(g) on a function f,analytic in ZI and t , can be obtained formally from (5.109) by setting Y = 0 and introducing the new variable 'u = z(z2 - 1)-ljZ. T h e result is [T(g)fl(v' t ,
v(l + 2bc) + abt + cdt-1 + bc) + abt + cdt-1][2vbc + abt + cdt-11 + ' at 1 + (c/al)2 + 2(c/at)v )'"], d (1 + bc)/a, (5.120) -i d 1 + (tb/d)2+ 2(tb/d)a =
[([2w(l
1)1/2
=
convergent for g in a sufficiently small neighborhood of e . As before, the functions B&) are the matrix elements of T(g) with respect to the analytic basis {f-,+,J:
c &c(g)f--u+c(u, t ) . m
[T(g)f-u+,l(v,
t)
=
z=o
204
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
valid
0,
-
for
I T~
I < I , I +yZ - 2T-lYV 1 0 the identity reduces to
- 2TV
1, -2 ,... . If y
=
By construction, C;(ZI)E 1, so for k well-known generating function (1
For v
=
=
< 1,
and
2p #
0 the above equation yields the W
+
T2 -
2T'UpP=
TccY(V). 1=0
0 it follows that (1
+
."-p
whence if
1 is odd,
if
1 = 2k is even.
Another useful relation for the Gegenbauer polynomials can be obtained by setting -7 = -y/t in (5.121) and going to the limit as y + 0. T h e result is (1
+ t2 + 2tv)k'/2C;
If v = 0, x
[(I
=
t( 1
+ v +i 2t t V ) t2
1,2]
=
1=0
(2""I'
-
1) tzc;-z(V).
+ t 2 ) - 1 / 2 this , expression takes the form
5-13. TYPE A REALIZATIONS OF
Ju
205
or C$.(X) =
5-13
z=o
xy1
-
xy-1
(2p + 212k
-
1 ) ( Z l ) j
Type A Realizations of J,
Fundamental identities for the Jacobi polynomials can be derived from realizations of J, , 2u not a nonnegative integer, by the type A operators (5.1 12). According to the usual procedure, in order to construct such realizations we must find basis vectors fm(z,t ) = Z,(z) t m ; m = u - k, k = 0, 1, 2 ,..., such that
(41
d2
- 2) @
+ [-2u + z ( k + 4 + u
-
d
l)] dx
-
k(g
+ u ) ) Z,-,(X)
= 0.
These relations determine the functions Zupk(z), k 3 0, to within an arbitrary constant. Indeed, the relation z dldz ZJz) = 0 implies ZJz) is a constant function. If we set ZJz) E 1 and use the second of Eqs. (5.122) to define the functions Z,-,(z) recursively, we easily obtain the result that the eigenfunctions are Jacobi polynomials, Z,-,(Z)
=F(-k,
--q
-~ U;
- 2 ~ ; z),
k
=
0 , 1, 2,...,
and that these polynomials satisfy all of the relations (5.122). T h e Lie derivatives (5.1 12) induce a local multiplier representation T‘ on With respect to the analytic basis {fir-Jz,t ) = Z,_,(z) tu-k} computed here, the operators T’(g) defined by (5.1 13) have matrix elements C,,(g), (5.31):
a.
206
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
This expression is equivalent to the identity (1 $-yT-')'((l
c
-
(Z -
r(-v Tz--l:
z=o *
-
r(+
l)T)-p(l
-
k
+
T)ll-'-'
+ 1) F ( - k , v + I ; I k + 1;y) r(I k + 1) -
I + 1)
-
F(F-2,CL; v ; X I , l ( ~ - l ) ~ l < l , l ~ l < l ,k = O , 1 , 2 ,...,
(5.123)
valid for all p, v E @ such that v is not a nonpositive integer. For y this identity simplifies to
=
f
l=O
Tz(-
=
0,
V-k ) F ( - k - 1, p ; V ; X), I l(X-1)Tl
< 1,
171
< 1.
I n particular if K = 0 we have a simple generating function for the hypergeometric polynomials: (1
- (X -
l ) ~ ) - p (+ l T)P-"
m
=
C
T'
(;')F(-Z,
p ; v ; z).
(5.124)
1=0
Finally, if we set y
= WT
in (5.123) and take the limit as T 4 0, we obtain
-k + I , p ; v ; x).
5-14
Type A Realizations of D(2u)
We shall now construct realizations of the finite-dimensional representation D(2u), 2u a nonnegative integer, in terms of the generalized Lie derivatives (5.106). I n order to compare the results more easily with the computations in Section 5-16, we choose basis vectors p,,, , m = - u , -u + I , ..., u - 1, u, for the representation space such that the action of the infinitesimal operators is
J3pm= mp, ,
J-P,
=
[(u
-
m
J+p,
= [(u -
m)(u
+ l)(u + ~~)11'2p,-1,
+ m + 1)71'2~m+l I
C0,,p,,
= u(u
+ llPm,
5-14. TYPE A REALIZATIONS OF D(2u)
207
see (5.41). Hence, to construct a realization of D(2u) with the operators
u, such we must find basis vectors p,(z, t ) = Wm(z)tm, m = -u,..., that .4 (z)ucA4
(aIl(1 - Z.1
+
d l -z u1z
P
4 + zp
zir I
(
-
,+)
(We assume the functions W,(z) are defined in the z-plane cut along the real axis from - 00 to 1.) These relations determine the W,(z) to within a multiplicative constant. I n fact, from the first of the above equations we have
+
whence
where c, is a constant. Using the second of Eqs. (5.125) we can define the functions W,(z) recursively from W,(z). A straightforward computation gives
5. LIE THEORY A N D HYPERGEOMETRIC FUNCTIONS
208 for m
J-p-,
= -u, = 0 or
-u
+ 1,..., u. However, we must still satisfy the condition
According to the first expression (5.126), this condition can be satisfied if and only if u f r are both nonnegative integers, i.e., if and only if r = -u, -u I , ..., u. For convenience we fix the constant c, by the requirement c, = ( - 1 ) p - r + r ) ! (2u)!]112
+
(u
+ r)!
Thus, W,(z)
=
(-l)?n+l. [;u
u
[(”
(u
- r)!
+- m)! m)! ( u + r ) !y2 By+), (u - r ) ! r , m = -u,-u+
1,..., u.
T h e normalization constant c, has been chosen so that the eigenfunctions W J z ) will be readily comparable with the matrix elements of irreducible representations of S U(2) (Section 5-16). From the commutation relations of the operators J*, J3, it is easy to check that the functions Wm(x), m = -u,..., u, satisfy all of the equations (5.125). Thus, the vectors p,(z, t ) = WnL(z) tm form a basis for a realization of the representation D(224). As we have shown earlier, the type A operators (5.106) determine a local multiplier representation T of SL(2) in terms of operators T(g),g E SL(2), defined by Eq. (5.109). With respect to the basis vectors p,(z, t ) , the matrix elements of this representation are Q l k ( g ) , Eqs. (5.43), (5.37). Therefore,
The following consequences of this identity are easily derived: If a = d = t = 1, b = 0, then
Rez
> 1, 1 c2(z + l)/(z - 1) 1 < 1, k
= 0,
1,..., 2u;
OF D(2u)
5-14. TYPE A REALIZATIONS
while if a = d there follows
Re
=
cosh(w/2), b
> 1,
=c =
sinh(w/2), p
=
209
cosh w , t
> 1, 1 ( p - 1)(z + l)(p + l)-l(z a real, m = -u, -u + 1,..., u. Rep
-
= eim,
1)-1 I
< 1,
I n terms of the functions P;"(x) defined on the real interval x 1 by P>m(x) = e--in[(m+r) /21B37'm (x i0)(see (5.111)) (5.127) implies the relations -1
+
< <
0 (1 --stan-)2
(r-u+k)/2
0 (1 +Tcot-)2
0
. (1 + eis tan - cot 2
- P~"(cosB cos p =
1 2u
p;-t,m(COS
2
-
)
(-T-U+k)/l
0
PT.-"+k(COS
(r-m)/2
(1 - eia tan
2
tan
-T
sin 0)
3
(-T-m)/2
2
sin 8 sin p cos a)
p)p;'-U+z(COSB)&u-m+t)s
1=0
0
< p, 0 < rr,
OL
real, m
= -u
, -U
+ l )
..., +u.
I n all of these equations 2u is a nonnegative integer and r --u 1,..., u.
+
= -u,
210
5-15
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
Weisner's Method for Type A Operators
T h e identities for the hypergeometric functions derived so far have all been related to local multiplier representations of SL(2). We can follow Weisner [I], however, and use the operators (5.1 12) to derive identities which are not associated with local representations. T h e basic fact to be noted is: If f ( z , t ) is an eigenfunction of the operator Cl,o, i.e., C,,,f(z, t ) = u(u l)f(., t ) , where
+
C1,"
=
z2(1
-
2-+ z[-2u + z(q + 2u
a2
z ) - - z2t 822 at az
zt
.f [@T bt)(at
1-
c
-
c(z
-
This is true because the operators (J*),
1)) '
J3
a
l)] -
ax
Eq. (5.113).
commute with
+ J3J3 -
Ci,o = J+J-
+ at
-
J37
and is valid for all g E SL(2),and all (x,t ) in the domain off such that [T'(g)f](z, t ) can be defined. I t is not necessary that the convergence conditions following (5.1 13) be satisfied. Moreover, if the functionf has a convergent expansion of the form f(2,
t) =
c h,,(z)t" m
then h,(z) must be a solution of the hypergeometric equation d2 dz2
[z(I - z ) -
+ [-2u
-
z(m - q
-
2u
d + l)] -(u dz
-
m)(q
+ u)] hm(z)= 0.
Hence, if 2u is not an integer, h,,,(z) is a linear combination of the linearly independent solutions F ( m - u, - u - q; -2u; Z ) and u 1, u - q 1; 2 224;z ) . If h,,(z) is regular at z = 0 it is a multiple of the first of these solutions. As an application of these remarks we apply the operator T'(g) to the basis function f,,((z,t ) = F ( m - u, - u - q ; -224; z ) tm and cons-
+ +
+
+
211
5-15. WEISNER'S METHOD FOR TYPE A OPERATORS
ider the resulting expression in the domain where over a neighborhood of zero:
- u, -u
-
q; -2u;
T =
t-l # 0 ranges
27
ab(1 - C T ( Z
-
l)/U)(1 -1(1
+
'1
bC)T/Ub)
TPu
(5.128) valid for ub
U
if c # 0. T h e expansion on the right-hand side of this expression follows because the left-hand side is equal to T - ~times a function of T analytic at T = 0. Since 1 T I < 1 ab/(l bc)l the group element g is bounded away from e and we cannot use the representation theory of SL(2) to obtain the expansion coefficients hu+, . However, these coefficients can be obtained directly. T h e left-hand side of (5.128) is regular at z = 0, which implies
+
h,-,(g, z ) = hA(g)F(-n, -u - q; -224; z ) ,
=
0, 1 , 2,... .
Setting z = 0 in (5.128) and comparing the resulting expansion with the generating function (5.124) for Jacobi polynomials, we have h i ( g ) = au+m-nbU-"cfiF(-n,
m
-
u ; -2u;
-(bc)-l)
(2un ).
There are two interesting consequences of this computation: If a = b = 1, c = 0 then IzA(g)
and
=
("-")n
212
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
where p = m - u, p = -u - q, v we have the symmetric relation
=
-2u. For a
=c =
1, b
=
-w-l
This is a bilinear generating function for the Jacobi polynomials. As another example consider a solution f(x, t ) of the simultaneous equations
+
C1,o.f = 4~ l ) f ,
Jf'
-f,
which is regular at z = 0. Thus, f satisfies the equations
Equation (ii) has the general solution f ( z , t ) == tU exp(t-l) h(zt-l) where h is arbitrary. Substituting this expression into (i), we find h(x) satisfies the confluent hypergeometric equation x
d2h dh + (2u - x) dx
dx2
(A. 11). Since f is regular at x multiplicative constant)
=
+ ( 4 + u)h = 0,
0, the solution is (unique to within a
f ( z ,t ) = U t exp(t-1) lFl(q
T
+ u ;2u; zt-l).
Note that t-uf(z, t ) has a power series expansion in 7 = t-l about 0. Similarly [T'(g)f](z, T - ~ ) is a solution of the equation
=
5-15. WEISNER'S METHOD FOR TYPE A OPERATORS
C,,JF(g)f]= u(u + l)[T'(g)f] and has an expansion in coefficients are multiples of Jacobi polynomials:
W'(g)fl(z, T-l) =
r-u(a 1 F l
- c(z
[q
+
+
- l)r>*++"(a
u ; 2%
CT)-*+~
exp
27
( a - c(z - 1)7)(a
+
ab
[ C).
1
+ +
213 7
whose
1
(1 ~ C ) T a(a + cr)
m
=
1jupn(g)F(--n,
-u
-
q; -224;
Z)T++",
n=O
/TI
<min
(I
a
___ c(z - 1)
I. I c I) . a
(5.129)
We can compute the coefficients j,-,(g) by setting x = 0 and comparing the resulting expression with the generating function (5.101) for the generalized Laguerre polynomials: j u-n (g) = &@/a( -c/a)nLA-2u-l) (1 iac).
For the special case b whence
=c =
0, a
= 1, this result
simplifies to
ju-,(e) = (rz!)-l,
If a
=c =
w-112,
b
=
0, we obtain
m
=
1 (-T)nLt-l'(W)F(--n, p;
V ; Z),
7L=O
I T I < min(1, I z
-
l)-l).
Finally, we can make sense of (5.129) even as a 4 0 if we assume 1, (1 + bc)/a = d+O: zt t = T-l. t r y 1 - z)g+uet1 ~ (q 1 + u ; 2u; 1--z
c = -b-+
-j,
This function has an expansion in terms of t-u+n, n 3 0, which leads to
z f 1.
(5.131)
214
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
T h e above expression can also be easily obtained from (5.130) and the transformation formula F(f 3 p ; v ; z ) = (1
-
z)pF
("
- p,
p; v ;
5 1
(A. 8).
Viewed another way, (5.130) and (5.131) constitute a proof of this transformation formula. Group theoretic methods can be used to derive many more such identities which are equivalent, by means of a transformation formula (A.8), to identities already given, but this will not be carried out here.
5-16
The Group SU(2)
At this point we begin a study of the representation theory of real 3-parameter Lie groups whose Lie algebras are real forms of 4 2 ) . Our treatment will be brief since the representation theory of these groups is well known and since many of the results for special functions obtained from such a study are special cases of results derived earlier in this chapter. SU(2) is the group of all 2 x 2 unitary unimodular matrices, i.e., the group of all matrices of the form (5.132)
where a, 6 E IT and 5 is the complex conjugate of a (Hamerniesh [l], Chapter 9). T h e identity element of S U ( 2 ) is the 2 x 2 identity matrix and the inverse of a group element is its conjugate transpose:
Clearly SU(2) is a real 3-parameter Lie group; furthermore, it is a subgroup of SL(2). T h e Lie algebra 4 2 ) of S U ( 2 ) can be identified with the space of all 2 x 2 complex skew-Hermitian matrices of trace zero. I n particular, as a basis for 4 2 ) we can choose the elements
with commutation relations
5-16. THE GROUP SU(2)
215
where [ a , 81 = 018 - @a for 01, ,9 E $ 4 2 ) . Comparing these commutation relations with expressions (2.6), Chapter 2, we see that 4 2 ) is isomorphic to the Lie algebra L(0,) of the 3 x 3 rotation group. T h e isomorphism of su(2) and L(0,) can be extended to a local isomorphism of the corresponding groups. I n fact SU(2) is the simply connected covering group of 0, and there is a group homomorphism R of SU(2) onto 0, such that A
(i
x p j ) A-l =
j=1
where x
=
3=1
(xl,x2, x,) is a column vector, A u --
( 0 1) 1 0’
2
=
0
(i
(5.135)
(R(A)x)$
oz)’
-’
E SU(2),
(13 =
(-10
j
0 1
are the Pauli matrices, and R(A)E 0, (see Gel’fand et al. 113). This equation defines the 3 x 3 matrix R(A)uniquely. Here R(A) = R(--A) so the homomorphism is not an isomorphism. T h e 2 x 2 matrices y+,2-, 9, defined in terms of t h e matrices (5.133) by
$*
=
F A + iJ%,
=
f 3
iA
satisfy the commutation relations
[43,$*I
= It$*,
I$+, P-I = 2Y3
and generate a complex Lie algebra isomorphic to 4 2 ) . Thus, the complexification of 4 2 ) is isomorphic to $42) and 4 2 ) is a real form of 4 2 ) . It follows from this relationship between the two Lie algebras that the abstract irreducible representations D(u, m,,), , 4, , D(2u), of 4 2 ) induce irreducible representations of 4 2 ) . We shall determine which of these induced representations of 4 2 ) can be extended to a unitary irreducible representation of SU(2). T o proceed with this determination we follow the technique initiated in Section 3-6. Consider a unitary irreducible representation U of SU(2) on a Hilbert space 2 and define the infinitesimal operators J1, J 2 , J3 by
r,
JJ
=
d d t ~ ( e x ptyk)ji , f-0
R
=
1,2,3,
(5.136)
216
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
for all f E 9. (9is a dense subspace of X which satisfies the properties (3.45), (3.46).) By definition, on the domain 9 these operators satisfy the commutation relations
Hence, the operators J*, satisfy the relations
defined by
J3
[J3, J'I
=
Jh =
[J+, J-I
*J*,
J,
+ iJ,,
=
J3
iJ3
= 2J3
and determine a representation p of the complex Lie algebra sZ(2) on 9. T o begin with, we shall investigate which of the finite-dimensional representations D(2u), 221 a nonnegative integer, can be obtained in this manner from a unitary representation U of SU(2). That is, we shall determine the conditions under which p can be identified with D(2u). I n this case X is finite-dimensional so 9 = X . From Lemma 3.1 we require h) =
(Jkf,
-
-(f, J&>,
=
1,2,3,
for allf, h E X, where ( -, ) is the inner product on X. In terms of the operators J*, J 3 these conditions can be expressed as
(J3f,
h)
=
(J+f,
(f,J3h>,
h)
=
(f,J-h).
(5.137)
T h e (2u + 1)-dimensional representation D(2u) is determined by the conditions JPm =
mPm
9
J+Pm
= [( u - m)(u
+ m)(u
+ m + l)li'2Pm+i
,
+ 1)11~2~m-l where m ranges over the spectrum S = { - - u , --u + 1,..., + u} J-P,
= [(u
-
m
(5.138)
on the left-hand sides of these equations. (For convenience the basis vectors p , have been normalized in accordance with (5.41) rather than (2.27).) As usual we apply conditions (5.137) to the operators J*, J3. Thus, m ( ~ m Pn) =
(J3pm
> Pn)
= (Pm
7
J3Pn> = n(Pm
9
Pn),
or (m - n ) ( p , ,p , ) = 0 for all m,n E S. This implies (prrL,p,) for m # n. T h e relation
=
0
5-16. THE GROUP SU(2)
217
implies 1 p,, 1 = 1 p , j for all m , n E 5. Without loss of generality we can assume that the basis vectors are of length 1. Thus, conditions (5.137) merely require that the vectors p m form an orthonormal basis for 3Ep and in no way restrict the value of the nonnegative integer 2u. Conversely, we will show that every finite-dimensional representation (5.138) induces an irreducible unitary representation U of SU(2). First, every element A of SU(2) can be written in the form A
=
(exp %23)(exp eA)(exPP)Ze83) O )
e+i~2/2
e+(Pi+d/Z - (;,+icm 1 - 9 2 ) P
cos(0/2) sin(@/2)
1.
ie-i(Pi-4P sin(0/2)
(5.139)
e+i(mi+oz)l2 C O S ( @ / ~ )
I n fact, if A E S U ( 2 ) is given by (5.132) the parameters rpl , 8, 'p, can be defined uniquely by cos(012) = 1 a 1,
sin(0/2)
=
1 6 1,
0
<0 <
T,
and
whenever ab # 0. If ab = 0, an infinite number of values of rpl, rp2 will satisfy (5.139). Expression (5.139) shows that the elements of SU(2) are completely determined by the elements fl , Y2, y 3of $ 4 2 ) ;hence, that the operators U(A), A E SU(2) are determined by the infinitesimal operators J 1 , J ,, J 3 , Eq. (5.136). Thus, the irreducible representation (5.138) of su (2) uniquely determines the cnitary representation U of S u ( 2 ) from which it is derived. T h e actual construction of the unitary irreducible representations of SU(2) is easy; we merely restrict the representations D(2u) of SL(2) to SU(2). For every nonnegative integer 2u, denote by Xu the Hilbert space consisting of all complex linear combinations of the basis vectors p,?,(z)= ( - z ) u + m / [ ( u m ) !( u - m)!]1/2;m E S = { - u , -u 1, ..., +u}, with inner product determined by ( p , ,p,> = a,,,,, , m, n E S. T h e elements of Xuare polynomials of order <2u in the complex variable z. We construct a representation D, of S U ( 2 ) on Xu in terms of linear operators Uu(A):
+
+
(5.140)
218
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
for a l l f e X u .For A E SU(2), (5.132), the operator D(g) given by (5.34) is formally identical with Uu(A). Th u s we immediately obtain the relation U"(Al)U"(A,)
=
UU(A,A2),
A, A, E S U ( 2 ) , 9
which proves D, is a representation of SU(2). T h e infinitesimal operators of of this representation corresponding to the elements $; , y2,j3 su(2) are given by J - i u z + - - -i - zd2 - i d 1 2 dz 2 dz' I d 1 d J -uz-----zz2 2dz 2 dz' . d J, = iu - zz -. dz Applying these operators to the basis vectors p,,(z), m (Jkf,
h)
=
-(f, J h ) ,
=
E
S, we find
1,2,3,
for all f,h E Xu. Thus, ((exp OJk)f, h ) = (f,(exp -OJk) h ) where exp eJk = O l ( Jk)l/Z! . (Since X, is finite-dimensional the operators Jk are bounded and there are no convergence difficulties.) This proves that the operators exp OJ,, , k = 1, 2, 3, on Xu are unitary. Moreover, we have
xF=o
Uu(A) = exp(ViJ,) ex~(OJi) ~XP(F,J,)
where A unitary.
E
SU(2) is given by (5.139). Hence, the operator U"(A) is
Theorem 5.3 D, is a unitary representation of SU(2) for all nonnegative integers 2u. T o compute the matrix elements
U&(A)
=
n, m
(P, , U"(A)p,),
E
apply Uu(A) to the basis vector p , :
c U,,(AIPn U
U"(AlP,
Thus,
=
n=-u
S, A
E
Su(2),
5-16. THE GROUP SU(2)
21 9
From this formula it is easy to derive the symmetrical expression
If this last expression is multiplied by sZu and summed over all nonnegative integers 2u one obtains the generating function Q(A;s, w ,z )
= exp s[(bz
+ 6)+ w(az - 6)]
Explicitly, the matrix elements are given by
1 r ( m-n
+ 1)F ( - u
-
n ;m
-
u; m
-
n
+ 1; -1
b/a 12)
where P>"(x) = e--i?r[(m+r)/21Bj:,m(x + iO), and the coordinates vl,8, y z for A are defined by (5.139). T h e addition theorem for the matrix elements is U
U,m(AIA,)=
1 U;j(A,)U;m(Az),
n, m
= -u
)...)u.
7=-u
We shall not work out the implications of this identity for special functions since all of the results are merely special cases of Eqs. (5.39). T h e fact that D, is unitary does lead to additional information, however. Indeed, U;#-1)
=
which implies (-l)m-np-n,nl (cos
Furthermore, I U,",(A)I
0)
=
< 1, or
U&(4
(u
+ n)! (u - m)!
(u
-
n)! ( u
+ m)!
P;m.n(COS
0).
220
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
T h e Haar measure on SU(2), suitably normalized, is given by d A = (1/16n2) sin 0 dyl d0 dy, where y1 , 8, y z , range over the values 0 8 T, 0 yl < 47r, 0 y 2 < 27r (Gel’fand et al. [l]). T h e matrix elements U,”,(A) satisfy orthogonality relations with respect to this measure. T o compute these relations we evaluate the integral
< <
Jr1;s:”
1 164
<
<
Q(A;s‘, 3 ,z’,)Q(A;s, w ,z ) sin 6’ dvl d6’ d y z
where the function Q( - ) is defined by (5.142). T h e integral on the left-hand side of this equation can be evaluated explicitly by power series expansion with the result
Thus, we obtain the relations
jm U n U m ( A )d A
=
8q.n
87.m 8w.u
2u+1
-
(5.144)
I n terms of the functions P ~ * m ( c 0) o s the orthogonality relations are
1;P ; ’ m ( ~f3)P;’m(cos ~~ 6’) sin 6’ d6’
2
+ 1s
= ___
2u
(u -n)! (u n)! (u
y u
+
-
m)! ,
+ m)!
‘
NOTE Since SU(2) is a compact group, these orthogonality relations could also have been derived from the Peter-Weyl theorem (Pontrjagin [l], Chapter 4). Moreover, the Peter-Weyl theorem can be used to I, ..., u ; show that the matrix elements {Uzn(A)};n, m = -u, -u 2u = 0, 1, 2, ...; form an orthogonal basis for the Hilbert space of all complex functions on the group manifold SU(2), square integrable with respect to the Haar measure d A . T h e irreducibility of the representation D, can be demonstrated exactly as in the proof of Lemma 3.2.
+
Lemma 5.8 D, is irreducible. T h e procedure for reducing the tensor product representation
Du 0D, of SU(2) into a direct sum of irreducible representations is
almost word-for-word the same as the procedure carried out for repres-
5-17. THE GROUP G,
221
entations of SL(2)in Section 5-5. This follows from the fact that the restriction of the representation D(2u) of SL(2)to SU(2)is the irreducible representation D, . I n particular we obtain D, @ D,
zminh.~)
@ D,+,-,
,
k an integer.
k=O
T h e Clebsch-Gordan coefficients C( . ) for this decomposition are given by (5.57) and (5.58). Thus, we find U t V
U;s(A)U;m(A) =
C
Z=lu-vl
C ( v , r ; u, n I 4 T
+ n)
. C ( v ,s; u, m I I, s + m)U,!+n.s+m(A),
which is a special case of (5.59). So far we have examined only those unitary irreducible representations of SU(2) which are induced by the representations D (2u) of sZ(2). We still have to consider the possibility of unitary representations of SU(2) induced by the infinite-dimensional representations t, , I,, D(u, m,) of 4 2 ) . However, as the reader can verify for himself, none of these infinite-dimensional representations induces a unitary representation of SU(2). I n fact, since SU(2) is a compact group, it follows from the Peter-Weyl theorem (Naimark [I], chapter 6) that all of its unitary irreducible representations are finite-dimensional. Furthermore, it can easily be shown that the representations D, , 2u a nonnegative integer, which we have constructed here are the only irreducible unitary representations of SU(2). For proofs of these remarks and a more detailed study of the representation theory of SU(2) the reader should consult the standard texts (Gelfand et al. [l], Hamermesh [l]).
5-17
The Group G,
G, is the real 3-parameter matrix group with elements
+
(The condition I a 12 - I b 12 = I means g has determinant 1.) These matrices do indeed form a group as is clear from the relation
222
5. LIE T H E O R Y AND HYPERGEOMETRIC FUNCTIONS
Since the elements of G3 are unimodular matrices it follows that G3 is a subgroup of SL(2). As real coordinates for G3 in a neighborhood of the x1 ix, , identity element e we can choose (xZ,y1 ,y,) where a = 1 b = y 1 iy, , x1 = - 1 (1 y: + y i - x : ) ~ / Using ~. the standard procedure for the computation of the Lie algebra of a local Lie group, we find L( G,) can be identified with the Lie algebra of matrices
+
+ +
+ +
Clearly, the elements 0
i/2
(5.145)
with commutation relations [$I
!
$121
=
-x
3
[$13
7
$11
= $12
[A 9
7
$121
=
-A
9
(5.146)
form a basis forL(G,). T h e exponential map fromL(G,) to G, has the properties
(5.147)
To give an explicit relation between L(G,) and sZ(2) we define 2 x 2 8,by $* = -yz & , Y3= i,$, . As is easily matrices B+,9-, seen, these matrices satisfy the commutation relations
[y+, Y-I =
~$13,
~
3
PI ,
=
u*;
hence, they generate a complex Lie algebra isomorphic to 4 2 ) . This shows that L(G,) is a real form of the complex Lie algebra 4 2 ) . T h e irreducible unitary representations of G, were first constructed by Bargmann [I], and this group has been the subject of numerous investigations ever since (Gel’fand et al. [2], Pukinszky [l], Vilenkin [3]). Here, we shall give a brief treatment of the representation theory.of G,
223
5-17.THE GROUP G,
to show the relationship between this theory and the identities for hypergeometric functions derived earlier in Chapter 5. T h e technique to be used should be familiar to the reader by now: Since L(G,) is a real form of sZ(2), the abstract irreducible representations D(u, m,), 7 , , 1, , D(2u) of $42) induce irreducible representations of L(G,). We shall determine which of these irreducible representations can be extended to an irreducible unitary representation of G, on a Hilbert space. It will turn out that all of the irreducible unitary representations of G, can be obtained in this way. Following the procedure introduced in Section 3-6, we consider a unitary irreducible representation U of G, on a Hilbert space X and define the infinitesimal operators Jk by (5.148)
for all f E 9. T h e 9 is a dense subspace of X satisfying properties (3.45), (3.46). On 9 we have
so the operators J*
= - Jz
[J+, J-I
f iJ, , J 3
=
2J3,
=
iJ3 satisfy the relations
[J3, J'I
=
*J*
and, thus, determine a representation p of 4 2 ) on 9. We shall first investigate under what conditions p could be isomorphic to the representation D(u, m,) on some dense subspace 9' of 9. Recall that u and m, are complex numbers such that m, f u are not integers, Re m, < 1. There is a basis i f m ) , m E S = (m, n: n an and 0 integer), for the representation space of D(u, m,) with the properties
+
<
J3fm =
mfm 7
Ci,ofm
J+fm = ( m - u>fm+l,
=
3-fm
=z
-(m
(J+J- + J3J3- J3)fm = 4 u + 1)fm
+ u>fm-~ *
Thus, if p is isomorphic to D(u, m,) on 9' we can assume {f,) is a basis D(-u - 1, m,), without loss for the space 9'. (Since D(u, m,) of generality we assume either I m u > 0 or I m u = 0, Re u >, -4.) From (5.147), exp 4 r y 3 = e. Th u s we must have U(exp 477y3) = exp 4nJ3 = I, where I is the identity operator on 2@. However, from the definition of the representation D(u, m,)it follows that (exp 477 J3)fm = e--4aimfn, for all m E S. This is possible only if 2m is an integer. Therefore, if D(u, m,) is induced by the representation U of G, then either m, = 0 or m, = +.
224
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
According to Lemma 3.1 the unitarity of U implies the conditions
(Jtf,
h)
=
-
1,2,3,
=
for all f,h E 9 where -)is the inner product on A?. I n terms of the operators J*, J3, the conditions become ( a ,
(JY
h)
(J+j, h )
(f, J3h),
1
=
-(f, J-h).
From the first of these relations we find
for all m,n implies
E
S. Th u s (fnL, fn)
(m - 1 - UKfm
,fm>
=
0 unless m
= <J?fm-l
,fm>
=
n. T h e second relation
=
-
=
(m
+ u)(
J-fm) fm-19 fm-1>,
or
+
= -8 iq, where q > 0. Substituting this value for u in the above relation we find 1 fnL-l I = I f, j for all m = 4 + n, n an integer. Thus, without loss of generality we can assume I fm I = 1. If m, = 0 the situation is more complicated. I n this case (5.149) implies either (i) u = -4 iq, q > 0; or (ii) u = q, -4 < q < 0. Corresponding to solution (i) we have I f m p1 j = ,fI I for all integers m, so that we can assume I f, 1 = 1. However, for solution (ii) ( m - q - l)ifm l2 = ( m q)Ifm-l 1 2 , all integers m. T o get an orthonormal basis for A?, define new vectors j, by
If m, = i,it follows that u
+
+
where we take the positive square root of the positive quantity inside the brackets. Then lj, I = lj,-l 1 and without loss of generality it can be assumed that I j, I = 1 for all integers m. At this point we have obtained the following possibilities for a representation of 4 2 ) on 9 induced by a unitary irreducible representation U of G3 on A?:
5-17. THE GROUP G,
225
(I) Aop* ( q > 0). There is an orthonormal basis {f,}, m E S (0, +l, f2 ,...}, for X such that JYm =
J-f m
=
mfm,
-(m - 3
+
=
J’fm +)fm-i
(m
+4
-
Ci,ofm
7
**
iq)fm+i, -(q2
+
t)fm
-
(5.151)
(11) A1/2,* ( q > 0). There is an orthonormal basis {f,}, m E S {&+, &$, ,...>,for X such that
JYm =
J-fm
mfm 7
1
-(m - 3
J’fm
=
(m
+ iq)fm-i
+B
- iq)fm+l
C,,ofm
-(q2
+
=
=
(5.1 52) t)fm
*
(111) AO,* (-4 < q < 0). There exists an orthonormal basis {j,), m E S = (0, i l , 1 2,...}, for X such that
J”j,
=
mjm ,
J-jm = --~,-,[(m where
E,
=
J+jm = 4
( m - q)(m
+ 4 + l)ll?L+l,
+ q)(m - 4 - l)11/2in-l ,
Cl,ojm = q(q
(5.153)
+ l)jm,
1 for ~ f>, l 0, e7,&= -1 form < O .
Conversely, in the following section it will be shown that each of the Lie algebra representations listed above does arise from an irreducible unitary representation of G, on R. Next we investigate under what conditions a representation 1, of 4 2 ) could be induced on 9’by a unitary representation of G3 on X . Here, 2u is not a nonnegative integer, S = {-u, -u 1 , --u 2,...I, and there is a basis {f,}, m E S , for the representation space 9’such that
+
JYm
=
mfm,
J+fm
=
( m - u)fm+l,
J-fm
= -(m
+
+ u>fm-i -
Exactly as in the treatment of the representation D(u, m,) given above, of sZ(2) to be induced on 9’by we find in order for the representation U it is necessary that (i) 2u is a negative integer; (ii) (fm ,f,) = 0 for m # n ; and (iii) (m - u)/(m u 1) = ,fI 12/jJn+l l2 > 0 for all m E S. This leads to the following possibilities: (IV) Of (n = 4, 1, #, ...). There is an orthonormal basis {j,}, m E S = {n, n + I , n + 2,...}, for %? such that
r,
+ +
J”j,
=
mjm,
J+jm = [(m
+ n)(m - n +
J-jm= - [ ( m - n)(m 4n - 1)]1/9m-1,
1)11’%+1
(5.154) Cl,ojm= n(n - l)jm
226
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
for all m E S on the left-hand sides of these equations. Here jm =
[(2n( m +l)!n( m l)!n)!]li2rm, u = -n. -
-
-
Similarly, the possible representations of the form by unitary representations of G, can be denoted by
1,
on 9’induced
(V) 0; (R = Q, 1 , $, ...). There is an orthonormal basis {jm), m E S = {-n, - R - 1, --n - 2,...}, for &‘ such that
JYm = mjm , J-j,
=
J+j,
[ ( m - n)(m
+n
= -
-[(m
+ n)(m
1)]1/7m-1 ,
-
n
+ 1)11’7m+l,
Cl,ojm = n(n - l)j,
(5.155)
.
Finally, as the reader can verify for himself, the only possible finitedimensional representation D(2u) of 4 2 ) on X induced by a unitary representation of G, is the trivial 1-dimensional representation D(0).
5-18
U n i t a r y Representations of G,
Now that we have classified all of the possible irreducible Lie algebra representations of 4 2 ) induced by unitary irreducible representations of G, , we will show, conversely, that each of these Lie algebra representations uniquely defines a unitary irreducible representation of G, . T h i s will prove to be an easy task. T o begin with, it is convenient to adopt a new coordinate system for the elements of G, . From the identity sinhp/2 eciwl2 sinhp/2 coshp/2)( 0 coshpI2
0 eiUI2)
(5.156)
valid for a = e-i(p+u)/z cash p / 2 ,
b
= ei(u-dl2
sinh p / 2 ,
p , v, p
real,
(5.157)
we can conclude that every g E G, can be expressed in the form g
=
exP(PA) exP(P2?) eXP(vA)*
(5.158)
I n fact, for every a, b E @ such that I a l2 - 1 b l 2 = 1, it is possible to find real numbers p, p, v satisfying (5.157). T h e coordinates (p, p, V) for G, are analogous to the complex coordinates 01, w,p for SL(2),Eqs. (5.11). Since G, is a subgroup of SL(2) we can readily verify: If g E G, has
5-18. UNITARY REPRESENTATIONS OF G,
227
coordinates ( p , p, v), i.e., if (5.158) is valid, then considered as an element of SL(2), g has coordinates a, w ,/3, where a = -ip, w = p ,
/? =
-iv.
Equations (5.157) do not determine the parameters p, p, v uniquely. p < co in which Indeed we can restrict p to take values in the range 0 case these parameters can be defined by
<
1a p =
j
=
coshp,
-arg a - arg b,
16 1
= sinhp,
v = arg 6
-
arg a.
As an immediate consequence of the validity of the decomposition (5.157) for all g E G, we have the result that an irreducible unitary representation U of G, is uniquely determined by the infinitesimal operators J3 and J z . Indeed, if g has coordinates ( p , p , v) then U(g> = ex~(pJ3)exp(pJ4 ~xP(vJ,).
This result shows that the Lie algebra representations listed in Section
5-16 uniquely determine the group representations U from which they are derived. We now proceed to the actual computation of the unitary irreducible representations of G3 . Since G, is a subgroup of SL(2) we will be able to construct these representations as restrictions to G, of corresponding local multiplier representations of SL(2).
(I) A O s q ( q > 0). T h e Lie algebra representation Ao,q,(5.151), of sZ(2) on 9’is isomorphic to the representation D(-& iq, 0) on the abstract vector space V . I n Section 5-1 it was shown that this abstract Lie algebra representation induced a local multiplier representation of SL(2) defined by operators A(g) acting on the space Gll of all functions analytic in a neighborhood of z = 1. With respect to the basis m = 0, & I , &2, ..., the matrix elements A:i*(g)of this multiplier representation restricted to G, are given by the equivalent expressions
+
Ifm},
+
1 b j 2 these where g = exp(p#.) exp(py2) exp(v#,). Since 1 a l2 = 1 matrix elements are defined for all g E G3 , not just in a neighborhood of
228
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
the identity element. T h e addition theorem, (5.8), for the matrix elements becomes (on restriction to G,)
c A;jQ(g,)AqiQ(gZ), 02
4iQ(g1g2)=
j=-m
g1 gz E G3 * t
(5.1 60)
I n particular, the addition theorem is valid over the entire group manifold G, , not just in a neighborhood of the identity. (This last remark follows from the facts: 1 b/a I < 1 for all g E G, , and m,,= 0.) From expressions (5.159) and the transformation formulas (A.8), it is easy to verify the equality A OZ,kg (
g ) = AE;*(g-l),
k , 1 integers, g E G,
.
(5.161)
This equality implies that the infinite matrix [A;iq(g)],- a < I, k < co,is unitary for all g E G, . Thus, we have constructed a represen-
tation of G, by unitary matrices. We can shed more light on the unitary property of the matrix elements by returning to Eqs. (5.151) which define the representation D(-$ + iq, 0) of 4 2 ) on the pre-Hilbert space 9’. T h e operators J*, J3 given there satisfy the relations
<J+f,
h) =
- ( f , J-h),
(JY, h )
=
J3h)
( f i
for all f,h E 9, where , -)is the inner product on of these relations we can obtain the identity ( a
= Ak,(exp -6f-), Azk(expby+)
H.From the first
b E @,
(5.162)
where the matrix elements A,,(g) of D(-& + ip, 0) are defined by (5.9). Indeed, if I 3 k, then
wheref, ,fk are elements of the orthonormal basis {f,} for H.If k > I, both sides of (5.162) are zero. Similarly, the relation involving J 3 implies the identity A,,(exp
a 9 3 ) = A,,(exp
&f3),
aE
@.
5-18. UNITARY REPRESENTATIONS OF
G,
229
If
is in a sufficiently small neighborhood of the identity element then g
=
exp(-[./.If-)
exp(-ab$+)
exp(ay3)
where eaj2 = a. Thus,
where
We have derived the identity
only for g in a sufficiently small neighborhood of the identity element. However, by analytic continuation its validity can be established for all g E SL(2) such that both sides of the equation are defined. I n particular, if g E G, we find j = g-l E G, and (5.163) reduces to the unitarity relation (5.16 1). Our results can be phrased in terms of unitary operators on the Hilbert space X . T h e vectors ( f m ) , m = 0,& l , *2,..., form an orthonormal basis for y14 and the elements of X are vectors f = Cg=-mc, f m , such c, l 2 < 00. Iff = x c mf m , h = C d mfrrLare elements of X . that the inner product ( f , h ) is given by ( f , h ) = F,dm. T h e operators U(g), g E G, , defined on X by
xz=-m ~
xz=-m
are obviously unitary and satisfy the representation property
230
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
These operators thus define a unitary representation AOjQ of G,. Furthermore, by proceeding exactly as in the proof of Lemma 3.2, we can show that AO'q is irreducible. (Recall that the proof depends on the fact that none of the matrix elements A:kq(g)is identically zero.) T h e realization of the representation Ao,Q given here is purely abstract. However, it is not difficult to construct concrete function space realizations. For example Aoaq can be defined on the Hilbert space L 2 ( y )of functionsf(eip) square integrable on the unit circle. T h e inner product is given by ( f,h )
=
(277-l
271 _ _ 0
f (eiq)h(eiq)d y ;
f,h E L2(y).
T h e functions fm(y) = eimp, m = 0, & I , f 2 , ..., form an orthonormal basis for L,(y). With every g E G, we associate the operator U(g):
(These operators can be obtained formally from (5.4) by setting z = eip, m, = 0, u = -4 iq.) Since I(aeiq h)/(beiq z)I = I , U(g) is well defined. T h e reader can verify for himself that the operators U(g) define a unitary representation of G, on L2(y) with matrix elements
+
+
+
where Ayiq(g)is given by (5.159). For more details see Bargmann [l]. It is not worthwhile to work out identities for the matrix elements A:;P(g)implied by the addition theorem (5.160) since these identities are merely special cases of (5.1 lo). Similarly, the differential relations obeyed by the matrix elements are special cases of the results of Section 5-7. However, the fact that the matrices [A:'*(g)]are unitary does give us one bit of new information: T h e matrix elements satisfy the inequality 1 A;ig(g)I 1 or
<
I, k integers, q > 0 ,
p
>, 0.
(11) A1l2,Q ( q > 0). From (5.152) it follows that the Lie algebra representation A1l2?qof 4 2 ) on 9' is associated with the abstract representation D(-4 + iq, 4). As in (I) above, this abstract Lie algebra representation induces a local multiplier representation of SL(2) defined
5-18. UNITARY REPRESENTATIONS OF
G3
231
a,
by operators A(g) acting on , Eq. (5.4). T h e matrix elements Ai/2*g(g) of the multiplier representation, where g is restricted to G, , are given by
F(-Zq
~
I , -iq
+ k + 1; k I + 1; l b / a r(k - E + 1) -
12)
I, k integers,
( 5.1 64)
where g = exp(p$Z,) exp(py.) exp(v9,). These matrix elements are defined for all g E G, , rather than just in a neighborhood of the identity. T h e addition theorem for the matrix elements, Eq. (5.8), becomes m
At?g(g,g,)
=
2
A:12*g(gl)A:.P*q(gz),
(5.165)
?=-a
valid for all g, ,g, E G, . From (5.164) and the transformation formulas (A.8) we can readily verify the equality A1/2-y(g) Zk
= A;f*g(g-l),
k, I integers, g E G3 ,
(5.1 66)
which proves that the infinite matrices [Ai/2*g(g)] are unitary. Moreover, just as in the proof of the identity (5.163) we can show that the matrix elements A,,(g) of the representation D(-$ iq, $) of 4 2 ) obey the relation
+
(5.167)
For g E G, this relation reduces to (5.166). T o construct the unitary using these matrices we consider a Hilbert space representation A112*q % with orthonormal basis {f,J, m = *$, fg, ... . Then the operators U(g),g E G, , defined on % by
*+,
form a unitary irreducible representation of G, .
232
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
T h e representation A112,qof G, can also be given a concrete realization in terms of unitary operators on the Hilbert space L,(y). This time we label the orthonormal basis vectors for L2(p) as f + + k ( v )= eik9. Corresponding to each g E G, we associate the operator U‘(g):
(This operator can be obtained formally from (5.4) by setting z = ei9 m = 4, u = iq.) T h e operators U’(g) define a unitary representation of G, on L2(q)with matrix elements 9
-+ +
0
where AtL2*n(g) is given by (5.164). For the details of the construction see Bargmann [l]. T h e addition theorem for the matrix elements is again a special case of (5.1lo), while the recursion relations obeyed by the matrix elements are obtained from Section 5-7. T h e unitarity condition(5.166) implies I AiL2*‘(g)I 1 or
<
(cosh p )
1<
1,
1, k integers, q
> 0,
(111) Ao*q (-8 < q < 0). T h e Lie algebra representation Ao**, (5.153), of sZ(2) on 9‘is isomorphic to the abstract representation D(q, 0) on the vector space V . With respect to the basis {fm}, m = 0, & 1, &2,..., the matrix elements of D(q, 0), restricted to G, , are given by
With respect to the basis { j m } ,m
=
0, f l , f 2 , ..., where
5-18. UNITARY REPRESENTATIONS OF G3
233
see (5.150), the matrix elements of B(q, 0) can be expressed in the equivalent forms
where g = exp(pf3) exp(pY2) exp(vg3) E G3 is given by (5.156). The addition theorem (5.8) becomes
valid for all g, ,g, E G, . From the expressions (5.168) for the matrix elements it is easy to verify the identity (5.170)
which proves that the matrices [AE;g(g)]are unitary. Moreover, if we mimic the derivation of the identity (5.163), case (I), we can derive the relation
€or the matrix elements of D(q, 0), where g E SL(2). When g E G3 this relation reduces to (5.170). Let %? be a Hilbert space with orthonormal basis {jm}, m = 0, f l , 1 2 ,... . It is easy to show that the operators U(g),
define a unitary irreducible representation of G, o n X .
234
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
T h e problem of constructing a concrete function space realization < q < 0 is somewhat more difficult of the representation Ao,q for than the previous cases and will not be treated here. See Bargmann [l] for details of the construction. Again the addition theorem and recursion relations for the matrix elements are special cases of equations (5.110), so it is pointless to re-derive them. T h e unitarity condition (5.170) implies
-+
(IV) 0,' (n = 4,1, 4 ,...). According to Eqs. (5.154) the Lie of 4 2 ) on 9' is isomorphic to the represenalgebra representation 0,' tation t-, on the abstract vector space V. This abstract representation of 4 2 ) induces a local multiplier representation of SL(2) defined by the operators B(g) acting on the space GZi of all functionsf(x) analytic in a neighborhood of z = 0, (5.18). With respect to the basis K = 0, 1, 2, ..., the matrix elements restricted to G, take the form
(see Eq. (5.23)). However, in terms of the orthonormal basis {jnt}for 9,
the matrix elements of B(g) are =
(2%+ k - l)! I! 1P [k! (2n $- 1 - I)!] B Z k ( d
(2%
+ 1 - l)! I!
(5.171)
where g = exp(p.,",) exp(p9J exp(vy3) E G, is given by (5.156). Since 2n is an integer, these matrix elements are well defined over the entire group manifold G, . T h e addition theorem is
valid for all g, , g,
E
G, .
5-18. UNITARY REPRESENTATIONS OF G,
235
T h e infinite matrix [BTk(g)],I, k >, 0, is unitary, ie.,
This property can be verified directly from the explicit expressions (5.171). Alternatively, one could mimic the proof of the identity (5.163) and show that the restrictions on the operators J*, J3 implied by Lemma 3.1 lead to the following relation for the matrix elements of t-, :
(2n + k
-
I)! I! B l k ( g )= (2n
__ + 2 - I)! k ! Blcl(j),
2, k 2 0, g E SL(2).
(5.174)
For g E G, this identity reduces to the unitarity condition (5.173). we can now construct the unitary Using the unitary matrices [B&(g)] representation 0,'of G, on a Hilbert space &? with orthonormal basis (j,,,],m = n, n 1, n 2,... . I n fact the operators U(g),gE G,, defined on X by
+
+
c m
u(g).jn+k = q k ( g ) j n t L,
k
= 0, 1, 2>-,
1=0
clearly form a unitary representation of G, . Exactly as in the proof of Lemma 3.2, one can show that 0,'is irreducible. T h e representation 0,'can also be given concrete function space realizations. We mention Bargmann's construction. T h e Hilbert space X z nn, = 3,1, 3, 2, ..., is the space of all functionsf(z) analytic on the open unit circle 4 in the complex plane, AZ' = {zi? < I}, such that 1-1 lim _ _
14n
if(.)
rr
l2(1 - x Z ) ~ -dx~ dy
< 00,
I > 1.
+
Here x = x iy and the integral is taken over A. T h e limit Z-+ 2n is essential only for n = 3;otherwise it is redundant. T h e inner product (., .>zn on Zzn is defined by
Set fn+k(z)= zk, k
=
0, I , 2, ... . By introducing polar coordinates
z = reis, it is easy to show
236
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
Therefore, f ( z ) = Ckm_o ckzk is an element of X z nif and only if
According to (5.175) the functions
in+&)
=
+
[(2n k - l ) ! p , (2n - l)! k!
k = 0 , 1 , 2 ,...)
form an orthonormal basis for YZZn(compare with Eqs. (5.154)). On H2, we define the operators U'(g) by [U'(g)f](.)
= (bz
uz + 6 g G , , .fE + V n(F+>), f E
% I
'
(5.176)
Formally these operators are identical with the multiplier representation (5.18), when g E G, . Expression (5.176) is well defined since
+
is T hus I(az 6),'(bx + a)/ < 1 if 1 x 1 < 1, and the manifold mapped into itself. T h e reader can verify that the operators U'(g) define a unitary representation of G, which is isomorphic to D',. With respect the matrix elements are to the orthonormal basis <jn+l
7
u'(g)jn+k)?n = B r k ( g ) .
For more details on this representation see Bargmann [l]. Identities for special functions implied by the addition theorem (5.172) are special cases of (5.116). T h e unitarity condition on the matrix elements yields the inequality (2n + k - l)! k! ]1'2 B3_~-z.n+'(cosh p) (2n + 1 - l)! I! l , k = 0 , 1 , 2 ,...) n = l2 , 1
3, I , $, ...). According to (5.155) the Lie algebra representation D; of 4 2 ) on 9' is isomorphic to JPn. I n Section 5-3 we showed that J-, induces a local multiplier representation of SL(2) defined by operators C(g) acting on the space of all functionsf(z) analytic in a neighborhood of z = 0, (5.28). With respect to the basis (V) D, (n =
237
5-18. UNITARY REPRESENTATIONS OF G,
(f-n--k), k = 0, 1 , 2, ..., the matrix elements of the operators C(g), for g restricted to G3 , are k ! F(-Z, 2n
qg) = GZa--ln--kbk--lI!
+k;k
I + 1; j b/a 12) + 1)
-
T(k - I
,
1, k 2 0,
Eq. (5.31). I n terms of the orthonormal basis {jnJ, for X,
the matrix elements of C ( g ) are (2n
+k
-
[k! (2n + 1
l)! I ! I)! I l l ZC d g )
-
(5.177)
Comparing these expressions with the matrix elements B;”,(g)of the representation D: , (5.171), we find
Thus, from Eqs. (5.172), (5.173) we immediately obtain the addition theorem
c c:xgl)c,”,gz), m
c;k(glgz)
=
j=O
g1 > gz E G3
I
and the unitarity relation
Exactly as in the treatment of the representation D: , we can show that the unitary matrices [CT;c(g)] define a unitary irreducible representation D; of G3 . This representation has a concrete realization on the Hilbert space X2,described in case (IV). To construct it define operators U”(g):
pJ’’(glfI(4
cfi
=
+b
2E & + a)FZnf(G)? A
7
fc
X
n
I
238
5. LIE THEORY AND HYPERGEOMETRIC FUNCTIONS
obtained formally from the multiplier representation (5.28) by restricting g to G,. As the reader can verify, these operators define a unitary irreducible representation of G, on X., , isomorphic to D;. With respect to the orthonormal basis
for
the matrix elements of these operators are (jL-L
f
U’’(g)j-,-k)Zn
=
C%(g).
As shown by Bargmann, the above five classes together with the trivial identity representation constitute all the irreducible unitary representations of G, . A study of the orthogonality and completeness relations obeyed by the matrix elements of the unitary irreducible representations with respect to the Haar measure on G, is beyond the scope of this book. For these results see Bargmann [l]. Likewise we will not consider the important problem of decomposing tensor products of unitary irreducible representations of G, into irreducible representations. T h e decomposition 0,’ 00,”E
cc
C0
D;+n’+k
k=O
is a special case of our results on the representation
t,
@
t,,
in Section
5.6. However, a study of all possible tensor products of representations in classes (I)-(V) is rather involved. Indeed there are few published
results on this problem which are explicit enough for application to special function theory (Pukhnszky [ 11, Romm[l]). 5-19
Contractions of 9(1,0)
I n Section 2-5, it was shown that the Lie algebra 3(0,0) F, @ (&) was a contraction of 9(1, 0) g 4 2 ) @ (8).Indeed, F, is a contraction of $42). Similarly 9 ( 0 , I ) was proved to be a contraction of 9(1,0). These relations between Lie algebras suggest that the matrix elements of irreducible representations of F, and 9 ( 0 , 1) may in some way be obtainable as limits of matrix elements of irreducible representations $42). This is the case, as will be seen from the following example. Here we follow Wigner and Inonu [l], and derive a relation between Jacobi polynomials and Bessel functions. According to Section 5- 16, the irreducible unitary representation D, of SU(2), 2u a nonnegative
5-19. CONTRACTIONS OF 3(1, 0)
239
integer, can be realized on the (2u + 1)-dimensional Hilbert space Xu consisting of all polynomials of order not greater than 2u in the complex variable z. T h e inner product (. , .) on Xu(conjugate linear in the first argument, linear in the second) is uniquely determined by the requirement
< P:
>
~3= 8m.n
< m,n < u,
-u
9
where
+
Thus, the 2u 1 vectors p 6 form an orthonormal basis for X u .T h e infinitesimal operators J k , k = 1,2, 3, on Xu are defined by
Ji
id - -i 2 2 d -iuz + 2 dz 2 d z ’ (5.179) . d dz
J3 = -iu - zz - ,
and obey the commutation relations
[ J i , J21
=
I n terms of the basis
J3 >
[J3,
JJ
[J,
= J2,
7
J31 =
J1.
(5.180)
{pk}, the matrix elements of these operators are
i 2 [(u -
--
n , m = -u
+ I)(u + n)]‘/’
, -u
+ I,.;.,
+u.
(5.181)
240
5. LIE THEORY A N D HYPERGEOMETRIC FUNCTIONS
According to Eq. (3.64), K, = -ip cos a ,
K,
K3 =
sin a ,
= -ip
-
d
&,
p
> 0,
01
real,
are infinitesimal operators induced by the irreducible representation ( p ) of E , on the Hilbert space X . Clearly, [Ki Kz1 = 0 , 9
[K3 KJ 9
=
K, ,
[K3, K,]
=
--K,,
(5.182)
and the infinitesimal operators generate a Lie algebra isomorphic to LE3 , a real form of T3.With respect to the orthonormal basis h,(a) = einm, f n = 0, 1, 2, ..., for X the matrix elements of these operators are K ) n , m = (hn
, K3hm)*
(K3n.m = (hn > K,hm)* = (hn
(JQn,m
9
= --in
z
s**m >
= - - P(Sn-1.m
5
2
+
sn+l,m),
(5.183)
K2hm)* = (Sn+1*, - Sn--l,m), n, m = 0, +1, 1 2,... .
T h e inner product (., .)* refers to the Hilbert space X. Corresponding to the parameter E > 0 we can define a new set of generators for the Lie algebra of SU(2): GI
= cJi,
Gz = .Jz
,
G3 =
J3 *
(5.184)
T h e structure constants for the commutation relations of the G operators are functions of E. Indeed,
As E -+ 0 the structure constants approach limits which are the structure constants of a new 3-dimensional Lie algebra. I n the limit the commutation relations become
which defines a Lie algebra isomorphic to c ? ~ . We have rederived the fact that €, is a contraction of 4 2 ) . Using this relationship we can obtain the irreducible representation (p) of LE3 as a limit of a sequence of representations of 4 2 ) . Consider the representation D, of SU(2),u a nonnegative integer. T h e matrix elements of the G operators corresponding to this representation can be deter-
5-19. CONTRACTIONS OF 9(1,0)
241
mined from (5.181) and (5.184). Computing the limit of these matrix elements as E -+0 and u -+ co in such a manner that EU -+p, we find
+
lim (Gfl')n,m= lim ( P P / ~ , E J, f-10
E"O
P
lim (G;'f)n,m = -2- (%+l," E+O
lim (G,P I C C-0
=
-in
pc)
- &-I*")
=
=
iP 2 (6n+1,m
--
=
(KP),,"
(Kp3),,, ,
n, m
+ 4-1.m)
= (K9n.m
?
(5.185)
7
= 0,
5 1 , -+2,...
Thus, we have obtained the irreducible representation ( p ) as a limit of irreducible representations of 4 2 ) . From this result it is easy to compute the effect of the limiting process on matrix elements of the representation 0,of SU(2). For our purpose it will be sufficient to consider the matrix elements of the unitary operator eeJl = Uu(exp 6yJ. From (5.143) we have [(" + m)!(u - n)!1'/2 (eeJi)n,m = (i)"-" ~ ; n y Ce).~ ~ (u - m)!(u n)!
+
Similarly, from (3.57) it follows that the matrix elements of the unitary operator eeK1,corresponding to the representation (p) of E, , are given by (eeKl)n,m = im-nJn-m(pt)).
T hus the first of the relations (5.185) implies lim (eeGl)n,m €40 ru-p
= (eeKl)n,m,
GI
= cJ1
,
(5.186)
or
Simplifying this expression we obtain lim (-k)"-"P~"*"(cos x / k ) = Jn-(x),
k++m
n, m
=
0 , & l , f2 ,..., xreal.
(5.187)
This demonstration of the limit relation (5.187) is not rigorous since the validity of (5.186) has not been explicitly verified. However, it is now easy t o establish (5.187) directly from the power series expansions for the functions involved. Using similar arguments we could derive formulas expressing Bessel functions as limits of matrix elements of irreducible representations of G3 . Also we could use the fact that 9(0,1) is a contraction of 9(I, 0) to obtain the associated Laguerre polynomials as limits of Jacobi polynomials. Both of these constructions are left to the reader.
CHAPTER
6 Special Functions Related to the Euclidean Group in 3-Space
I n this chapter type A and B operators forming a realization of 4 2 ) will be extended to type E and F operators, respectively, forming a realization of the 6-dimensional Lie algebra F,. T h e type E and F operators yield recursion relations for hypergeometric and confluent hypergeometric functions which are of a different nature than the recursion relations derived so far. In particular it is much more difficult to compute the matrix elements of multiplier representations induced by these operators. For this reason the results presented here are incomplete, and much remains to be done to obtain all of the special function ,. identities implied by the representation theory of F T h e Lie algebra of E, , the Euclidean group in 3-space, is a real form , . I n Sections 6-4 and 6-5 this relationship will be used to compute of F matrix elements of the unitary irreducible representations of E, . T h e matrix elements turn out to be spinor-valued solutions of the wave equation (V2 u * )Y(r) = 0 and are of wide applicability in theoretical physics. These results were first computed by Vilenkin et aZ. [I]. See also Miller [2].
+
6-1. REPRESENTATIONS OF
6-1
243
Representations of F6
9,is the &dimensional complex Lie algebra with generators P+,8-,8 3 , $+, $-, $3 and commutation relations
[Y3,61 '
=
I$+,
&Y*, [Y3,9*1= [P3, $'I P+I = [g-, 9 9
=
=
f9*,
[ 9 3 , ~ 3 1= 0,
P-1 = [Y+, 9-1= 293, 9-1 = 8. [93,9*] = [9+,
[y+, y-1 = 293,
[f+,
(6.1)
It is straightforward to verify that the Jacobi equality is satisfied, so the relations (6.1) actually define a Lie algebra. Clearly, the elements f + , 8-, generate a subalgebra of T6isomorphic to 4 2 ) . Indeed we will identify this subalgebra with 4 2 ) . T h e elements P+,8-,9 3 generate a 3-dimensional abelian subalgebra of F , which is also an ideal. Denote by T , the complex 6-parameter Lie group consisting of all elements {w, g},
x3
with group multiplication (w, R } W , g'> = {w
+ gw', gg')
where the plus sign (+) denotes vector addition in gw
= (a2a -
b2p
+ aby, -c2a + d2p - cdy, 2acol
(6.2)
c3and
- 2bdP
+ (bc + ~ d ) y ) .
I n particular the identity element of T6 is (0, e } where 0 = (0, 0, 0) and e is the identity element of SL(2);and the inverse of an element {w, g} is given by (w, g}-1
= { -g-lw,
g-I}.
T h e associative law can be verified directly. T h e set of all elements of the form (0, g}, g E SL(2),forms a subgroup of T , which can be identified with SL(2). Similarly the set of all elements of the form {w, e } , w E forms a subgroup of T, which can be identified with It is straightforward t o show that 9, is the Lie algebra of T, . Indeed the $-,f3generatethesubgroup SL(2)of T6, generatorscan bechosenso y+, while {w, e } = exp(&+ ,W783) for w = (a,p, y ) E Consid-
c3,
c3.
+
+
c3.
6. THE EUCLIDEAN GROUP
244
IN
3-SPACE
ered as Lie derivatives on the group manifold of T 6 ,the generators are given by the expressions
(see Section 3-2). (It follows from the general results of Chapter 1 that as a local Lie group, T6 is uniquely determined by the commutation relations (6.1). However, the global structure of the group is not uniquely determined and the global group (6.2) has been chosen for convenience). by Consider a complex vector space V and a representation p of F6 linear operators on V . Set
~(9’) = P*,
~(9’”) = P3, p($*)
=
J*,
p(Y3) = J3.
Then the linear operators P*, P3, J*, J 3 satisfy commutation relations on V entirely analogous to (6.1), where now [A,B] = AB - BA for operators A and B on V. We can define two operators on V which are of They are special importance for the representation theory of F6. p*p
= -p+p-
-
p3p3,
p* J
==
--( P+J-
+ P- J+)
-
P3J3.
(6.4)
I t is easy to verify the relations
for all 01 E F6 . Thus if p is an irreducible representation of F6 we would J to be multiples of the identity operator on V. expect P P and P By restricting an irreducible representation p of F6 to the subalgebra $42) we can obtain a representation plsZ(2) ( p restricted to 4 2 ) ) of 4 2 ) . Suppose plsZ(2) can be decomposed into a direct sum of irreducible representations of $42) as classified in Chapter 5. I n this case the operators P+, P-, P3 may “mix up” the representations of sZ(2) and map basis vectors corresponding to one subspace of V , invariant under sZ(2), into other subspaces. By finding a functional realization of such a representation p it may then be possible to derive differential recursion relations
6-1. REPRESENTATIONS OF F6
245
relating special functions associated with different irreducible representations of 4 2 ) . It is this possibility which is a principle motiva. As will be shown, by choosing approtion for the consideration of r, priate irreducible representations of 9,one can obtain recursion relations for hypergeometric and confluent hypergeometric functions which are distinct from those derived in Chapters 4 and 5. T o be specific we will classify all representations p of F6on the vector space V with the properties: (i) V = CueQ @ V , , where the summation ranges over a countable number of values of u, each value of u occurring at most once. plsZ(2) acts irreducibly on each subspace V , and coincides there with one of the irreducible representations D(u, mJ,t, , 4, , D(2u) classified in Theorem 2.3. (ii) p/sZ(2) has one of the possible forms:
= c 0q u , T J , p l s W = c 0T, P W ) = c 0I,? plsZ(2) = 0W W ,
(1) p l s 4 2 )
UEQ
(2) (3)
9
us0
UEQ
(4)
ueQ
where in each case the summation extends over the same values of u E Q as in (i). (iii) p is irreducible. (iv) P . P $ 0 on V . T h e results of this classification are as follows: T h e basis vectors of each subspace V, of V are denoted by where J 3 f :) = m fg)and the { fg)} for a fixed u form a canonical basis for an irreducible representation of sZ(2) as listed in Theorem 2.3. Q = (u: V , C V , V , f O}. Let w and q be arbitrary complex constants such that w # 0.
fz)
, satisfying conditions Theorem 6.1 Every representation p of F (i)-(iv) is isomorphic to a representation in the following list:
1 j 4. Th e spectrum Q = { - q + n: n an integer q)/sZ(2) takes the form ( j ) given by condition (ii). For j = 1,2q is not a nonnegative integer. I f j = 2 or 3,, 2q is not an integer. If j = 4 then -2q is a nonnegative integer. (a)
t j ( w , q),
3 O}. T h e
fj(w,
< <
6. THE EUCLIDEAN GROUP
246
IN 3-SPACE
<
(b) li(w, q), 1 < j 3. 2q is not a nonnegative integer. T h e spectrum Q = {q - n - 1: n an integer 3 O}. T h e &(w, q)/sZ(2) takes the form ( j ) given by condition (ii).
< <
3. Here q and u, are complex numbers (c) R j ( w ,q, uo), 1 j q, or 2u, is an integer. Re u, < 1 and none of u, such that 0 Q = {uo n: n an integer}. T h e Ri(w, q, u,)/sZ(2) takes the form ( j ) given by condition (ii).
+
<
(d) R,(w, 0, 0). j
=
1, q = u, = 0, Q
=
{n:n an integer).
Corresponding to each of these representations there is a basis for V , , u E Q, of the form { such that
fz)}
T h e representations listed above are not all distinct. In fact we have the isomorphisms Tl(w, q) g J1(w, -q - 1 ) for 2q not an integer, and %(a,q, uo) R,(w, 9, -u, - 1). T h e details of the proof of this theorem will not be given since they are somewhat tedious. T h e method of proof closely follows that of Naimark [2], Chapter 3. For our purposes it is enough to know that the listed representations in Theorem 6.1 actually are irreducible representations of Y6and this fact is easy to verify from the explicit expressions
=
(6.5).
247
6-1. REPRESENTATIONS OF
For each of the representations listed above we define operators X ( u , +) and X ( u , -) such that X ( U ,+) = P-J+
+ P3J3+ + l)P3 (U
+ up3
X ( u , -) = -P-J+ - P3J3
-
-
2 % J3 - wqI, U + l
(6.6)
2 J3 + q I , U
where I is the identity operator on V . If fg)is a basis vector for V , then Eqs. (6.5) imply X(u,
X(u,
+)fZ’ w ( u u-+ql + 1)fE+l)’ =
-)fZ)
=
4.
+ m)(u
-
U
m)(.
(6.7)
+ q )f2-l)
*
T h e range of values of u and m for which these expressions make sense depends on the particular representation of F, with which we are concerned. It is clear from (6.7) that X(u, +) and X ( u , -) can be considered as u-raising and u-lowering operators, respectively, in the same way that J+ and J- are m-raising and -lowering operators. However, there is a profound difference between these two sets of operators. I n particular the X ( u , &) are functions of u and are not members of the Lie , . This is in distinction to algebra generated by the elements p(a), a E F J*. Consequently, it is reasonable to expect that recursion relations for special functions obtained from (6.7) should be of a different nature from the recursion relations derived in Chapter 5. , as given by For the classification of irreducible representations of F 0 on V . I t is possible to find Theorem 6.1 we have assumed P . P additional irreducible representations of F, which satisfy conditions (i)-(iii), but instead of (iv), satisfy
+
(iv)’ P - P = 0 ,
P.J
+0
on
V.
Again, only the results of the classification will be given. Theorem 6.2 Every representation p of F, satisfying conditions u,), 1 j 3, (i)-(iv)’ is isomorphic to a representation of the form Rl(<, where and u, are complex constants such that # 0, 0 Re u, < 1 and 224, is not an integer. T h e spectrum Q = {u, n: n an integer}. I?;(<, u,)jsZ(2) takes the form (j)given by condition (ii). Corresponding
<
<
+
<
< <
248
6. THE EUCLIDEAN GROUP IN 3-SPACE
to each representation Ri((,uo) there is a basis { fg)}, u E Q, for V such that
+
P
5(u
+(224 m)(u + m + 1)u
.Pf2’= 0,
-
1)
(%-I) fm-1
p . Jjg) ==
’
5f
-
) ;
-
With the exception of the isomorphism I?;(<, u,) I?;((, -u, - l), all of the above irreducible representations are distinct. (Equations (6.8) can be obtained formally from (6.5) by setting q = - [ / w and passing to the limit as w + 0.) For each of the representations R;([,uo) listed above we define operators X’(u, +) and X’(u, -) on V such that X’(u, +) = P-J+
+ P3J3 + (U + l)P3
X ’ ( U ,-)
5 + - P3J3 + up3 - - J3 + 51.
=
-P-J
c
- __
u + l
J3 - [I,
(6.9)
U
As a consequence of relations (6.8), these operators satisfy X‘(u, +)f:’
=
X ( u , -) f p’ =
---fE”) -5 u f l 5(u
,
+ m)(u U
(6.10) -
m)
f 2-l)’
where the range of values of u and m for which the expressions make sense depends on the representation I?;([, uo). T h e X’(u, 4) can be
6-2. TYPE E OPERATORS
249
considered as raising and lowering operators for the index u. However, these operators differ greatly from the m-raising and -lowering operators J+, J- since they are functions of u and are not elements of the Lie algebra generated by the p(a), 01 E r 6 . We could continue the classification scheme of representations of T6 given above and determine all representations satisfying conditions (i), (ii), (iii) and
P . J = 0 on
(iv)" P - P = 0,
I/.
However, it turns out that for such representations we must have Pf = P- z P3 E 0. Th u s these representations reduce to irreducible representations of 4 2 ) which have already been classified in Chapter 5.
6-2
Type E Operators
We designate as type E the differential operators
P3
p.p
= wz,
= -w2,
P'
= *wt*l(z2
J = -wq,
(6.11)
- l)l/2,
w,
q E @,
w
# 0.
Clearly these generalized Lie derivatives satisfy the commutation relations (6.1) and thus generate a Lie algebra isomorphic to Y6. (In order to properly define the square roots occuring in (6.11) we assume z takes the values in the complex plane cut along the real axis from -CO to + I . If z > 1 we require (9- l)lI2 > 0. T h e operators (6.11) are defined for all t E e( except t = 0.) Here J*, J3 are just the type A operators (5.106). As will be shown, type E operators can be used to construct realizations of the representations of Y6listed in Theorem 6.1. However, they cannot be used to construct realizztions of the representations given by Theorem 6.2, since if P * P = u 2= 0 then p+ = p- = p3 = 0. To construct the representations listed in Theorem 6.1 we must find analytic functions f g ) ( z ,t ) = Zg)(z)tm such that
250
6 . THE EUCLIDEAN GROUP IN 3-SPACE
and
valid for all u E Q (and for all rn in the spectrum of the representation of sZ(2) corresponding to u ) on the left-hand sides of these equations. Moreover, relations (6.7) imply
Clearly, all of the above equations are independent of the nonzero parameter w. T o find solutions {Z$y)(z)] for these equations it is enough to solve (6.12) and (6.14). Indeed the first of Eqs. (6.13) can be obtained by adding the two expressions (6.14). T h e second and third equations are simple consequences of the first and the relations [P3, J*] = *Pi. With this simplification it is not difficult to find solutions. T h e functions
defined for all u E Q and all rn in the spectrum of the irreducible representation of 4 2 ) corresponding to u satisfy the above equations unless j = 2. (The fact that these functions satisfy Eqs. (6.12) follows directly from Chapter 5. Equations (6.14) can be verified through direct
6-2. TYPE E OPERATORS
251
substitution of the infinite series expansions (A.9) for B L * ’ ~ ( ZHowever ). for certain representations these functions are identically zero.) T h e special case j = 2 is left to the reader. T o be more precise we must specify the possible values of u, m, and q which can appear on the left-hand sides of (6.12)-(6.14), corresponding to each irreducible representation of F 6 .T h e results are (4
TAU, 4 )
l<j<4; w f O 2q not a nonnegative integer; j = 1: u = -q, -4 1, -4 2,...; m = m , , rn, & 1, rn, i2,...; where m, is a complex number such that q m, are not integers and 0 Re m, < 1. j = 2: 2q not an integer; u = -q, -q 1 , -4 2,...; m = -u , -u + I , - u + 2 ,....
+
+
<
+
j
=
3: 2q not an integer;
+
+
+
u = -q, -4 1, -4 2,...; m = u , u - l , u - 2 ,.... j = 4 : -2q a nonnegative integer; u = -q, -q 1,...; m = -u, -u + 1 ,..., +u.
+
< <
1 j 3 ; w # 0 ; 2qnot a nonnegative integer;
(b)
u = q - 1, q - 2, q - 3, ...; m = m , , m, & 1, m, 2, ...; where m, is a complex number such that q f m, are not integers and 0 Re m, < 1. j = 2: u = q - 1, q - 2, ...; m = -u,-u+ 1 , - u + 2 ,.... j = 3 : u = q - l , q - 2 ,...; m = u , u - l , u - 2 ,....
j
=
1:
<
(c)
1<j<3;
Rj(w, 9, uo),
0
< Re u, < 1 ;
j -
=
j
=
qE@,
u, -& q and 2u, are not integers; 1 : u = u , , u , ~ 1 , u o ~,... 2 ; m = m , f I , m o i 2 ,...; where rn, is a complex number such that u, f m, are not integers and 0 Re m, < 1. 2: u = u,, u, & 1, u, i 2,...; m = -u,-u+I,-u+2 ,.... 3: u = u,, u, 1, u, & 2,...; m = U, u - 1, u - 2 ,... .
<
j
w i 0 ;
252
6. THE EUCLIDEAN GROUP IN 3-SPACE
( 4 R,(w, 0, O),
j=1;
w f O ;
q=uo=o.
u = 0, i l , 1 2,...; m = m,, mo f 1, m, f 2,...; where m, E $2 such that 0 < Re m, < 1.
T h e functions (6.15) are, of course, not the only solutions of our set of recursion relations. T h e construction of a linearly independent set of solutions is left to the reader. I n conclusion, we have found realizations by type E operators of most of the representations listed in Theorem 6.1. T h e basis vectors for these realizations are the functions (6.15) obtained from Chapter 5 in connection with the realization of representations of 4 2 ) by type A operators. Our results yield new recursion relations for these basis functions. I n particular we have seen that the recursion relations (6.14) are of an entirely different nature from the recursion relations derived earlier in this book. Equations (6.14) do not correspond directly to a but rather to a representation of the universal representation of 9, enveloping algebra of Y6(Jacobson [l], Chapter 5). Thus, it is not possible to obtain addition theorems and generating functions from these relations using the simple procedure we have employed u p to now. I n this connection it is interesting to examine Truesdell’s procedure for deriving identities for special functions directly from their recursion relations (Truesdell [l]). A careful analysis of his method reveals that it is basically Lie algebraic in nature. Indeed, his method succeeds for all of the recursion relations derived in Chapters 3-5, but fails for the recursion relations (6.14). Our more detailed analysis provides an explanation for this failure. given above to T h e problem of extending the representations of 9, local multiplier representations of T , is a rather difficult one and we will not solve it here. T h e principal difficulties involve the verification that the functions fg’(z, t ) = Ziz)(z)tm form an analytic basis for the representation space and the computation of matrix elements of the wP3). Among the addition theorems which operators exp(xP+ yPcan be obtained from such an analysis are identities involving spherical Bessel functions and Gegenbauer polynomials. I n Section 6-4 we will solve a special case of this problem to compute the irreducible unitary . representations of the group E, whose Lie algebra is a real form of T6 Finally, any attempt to find realizations of representations of 9, by generalized Lie derivatives in one complex variable is doomed to failure. It will be proved in Chapter 8 that no such realization exists.
+
+
6-3. TYPE F OPERATORS
253
6-3 Type F Operators T h e type F differential operators are
P3
= 252-1,
P*P=O,
P*
=
P'J=-[,
125ti1.2-1,
(6.16)
O#[E@.
These operators satisfy the commutation relations (6. l), and generate Here J+, J-, and J3 are the type B a Lie algebra isomorphic to Y6. operators (5.85), where we have set t = e T , q = 4. Since P . P = 0 the generalized Lie derivatives (6.16) cannot be used to find realizations of listed in Theorem 6.1. However, they are well the representations of Y6 suited for the realization of representations classified in Theorem 6.2. T o construct realizations of R;(<, uo) using type F operators we must find nonzero analytic functions f g ) ( z ,t ) = Z g ) ( z )tm such that
(6.17)
and
254
6. THE EUCLIDEAN GROUP IN 3-SPACE
valid for all u E Q and all m in the spectrum of the representation of 4 2 ) corresponding to u on the left-hand sides of these equations. From (6.10) we have the relations
All of the above equations are independent of the parameter 5. To solve these equations it is enough to find solutions for (6.17) and (6.19). This can be seen by noting that the first of Eqs. (6.18) is just the sum of the two equations (6.19). T h e second and third of Eqs. (6.18) can be obtained from the relations [P3, J*] = &Pi. It is now easy to find a solution. According to Section 5-8 on type B operators, the functions Z‘”yz) nl
m-u-l(4’ i = 1
= ( -z)u+le-z/2~(2u+l)
(6.20)
and the functions
(6.21)
both satisfy Eqs. (6.17). Moreover, by using the power series expansion for the generalized Laguerre functions it is easy to verify that these functions also satisfy (6.19). Corresponding to each irreducible representation Ri([,u,) of F6 the following range of values for u and m can appear on the left-hand sides of Eqs. (6.17)-(6.19): 1<j<3; [#O; 22.4, not an integer.
I;
O
j = 1: u = u o , u o i 1 , u o i 2,...; m = m o , r n , ~ 1 , m , ~,...; 2 where m, is a complex number such that m, integers and 0 Re m, < 1.
<
u,
are not
j = 2 : u = u o , u o ~ 1 , u , j ,...; 2 m = - u , -u + 1 , - u + 2 j = 3 : u = u,, u, i 1, u, 5 2 ,...; m = u, u - 1, u - 2 ,... .
,....
Thus, using type F operators we have constructed realizations of all representations of F6 listed in Theorem 6.2. Either (6.20) or (6.21),
6-4. THE EUCLIDEAN GROUP
E,
255
obtained from Section 5-8 in the study of type B operators, can be used as basis vectors for these realizations. Among the new relations derived for the basis functions are (6.19), which do not have the simple Lie algebraic interpretation enjoyed by the other recursion relations derived in this section. As might be expected from our corresponding discussion of type E operators, Truesdell’s technique for obtaining special function identities from recursion relations fails when applied to (6.19). Equations (6.17)-(6.18) can also be solved in terms of generalized Laguerre functions when 2u is an integer, even though this case is not included in the spectrum Q of any of the representations Ri(5, uo)listed above. Indeed, except for the singular values u = -1, -+,O, we can always find solutions. However, since the equations become meaningless for the singular values of u, these solutions do not form a basis for a representation of F6 . T h e relatively difficult problem of computing matrix elements of the local multiplier representation of T6induced by the irreducible represenwill be omitted. tation Rj(<,uo) of F6
6-4
The Euclidean Group E,
E6 is the real 6-parameter Lie group consisting of all pairs (r, A ) where r = (rl , r 2 , r3) is a real column vector and A is an element of S U ( 2) .T h e group multiplication law is (r, A)(r’, A ’ ) = (r
+ R(A)r’,AA’)
(6.22)
where the plus sign (+) means vector addition, AA’ corresponds to multiplication in S U ( 2 ) , and R(A) is a real 3 x 3 orthogonal matrix defined by (5.135). If A=(
a
-
b
then R(A) has the explicit expression
ad+b6=1,
6. THE EUCLIDEAN GROUP IN 3-SPACE
256
Since S U ( 2 ) is simply connected, so is E,. I n fact E6 is the simply connected covering group of the Euclidean group in 3-space: the group of all pairs (r, R), r a real 3-vector, R E 0, , with group multiplication (r, R)(r’, R’)
=
(r
+ Rr‘, RR’).
T h e two-to-one onto homomorphism relating the two groups is (r, &A) -+ (r, R(A)),where R ( A ) = R(-A) is defined by (5.135). E, can be considered as a real subgroup of T , . Indeed, as the reader can easily verify, the collection of all elements {w, A}, where w = ( + ( - r 2 - irJ, $(r2- irJ, -ir,), rl , r 2 , r, real, and A E S U ( 2 ) , forms a subgroup of T , isomorphic to E6 . T h e isomorphism is given by (r, A ) t+ {w, A}, r = ( r l , r 2 , r3). T h e real 6-dimensional Lie algebra g6 corresponding to E6 is generated by elements # k , 9$ , k = 1, 2, 3, with commutation relations
[da; Yd 1
[A
=~jkl2l
1
9 k I
== c i k 8 ?
9
[q.92 = 8, 7
j , k, 2 = 1 , 2, 3,
(6.24)
+
where ei kl is the completely antisymmetric tensor such that cIz3 = 1. T h e ykform a basis for su(2), a subalgebra of g 6 . We choose these generators so that they are related to the finite group elements by (r, 4= exp(r1?
+
yz%
t y 3 9 3 exp v d 3 exp 3 2 , exp v z h
(6.25)
where r = ( r l , r 2 , r,) and y l , 19,y z are the Euler coordinates for A , (5.139). T h e formal elements 9*, g3,y*,,f3 defined in terms of the generators (6.24) by B*
=
~9~ + iPl,
9 3
= i9,,
f’
=
*A + i f l ,
f 3
= iY3
can easily be shown to satisfy the commutation relations (6.1) for the complex Lie algebra F6.Thus, we have explicitly determined g6as a real form of F6. According to this result the irreducible representations of 7,, classified in Section 6- 1, induce irreducible representations of 8,by restriction. We shall determine which of these induced representations of 8, can be extended to an irreducible unitary representation of EBon a Hilbert space. Following the usual procedure we let U be an irreducible unitary representation of E, on a Hilbert space % and define the infinitesimal operators J k ,P, , k = I , 2, 3, by
6-4. THE EUCLIDEAN GROUP E,
257
for all f~ 9. (9is a dense subspace of S satisfying properties (3.49, (3.46).) On 9 these infinitesimal operators satisfy the commutation relations
[J,
7
Jkl
=
E ~ ~ L J Z[J, p ~ 1== E~~J'z [P, t
j , k, E
9
Pk1
=
= 0,
1,2,3,
where e j k l is the completely antisymmetric tensor such that T h e operators J*, J3, P*, P3 defined by
J*
=
+Jz
+ iJ1,
J3 =
iJ3,
Pi-
=
+P,
+ iPl ,
E~~~
=
+ 1.
P3 = iP, (6.27)
satisfy commutation relations analogous to (6.1) and thus determine a representation p of the complex Lie algebra F6on 9. We shall investigate the conditions under which p (restricted to some dense subspace 9' of 9 ) is isomorphic to one of the irreducible representations of F6 listed in Theorems 6.1 and 6.2. From the study of the representation theory of S U(2) given in Section 5-16, all of the irreducible unitary representations of SU(2) are finitedimensional. Thus, the only irreducible representations of F6which could possibly induce unitary representations of E6 are those of the form f4(w, q), w # 0, -2q a nonnegative integer. We shall determine under what conditions a unitary irreducible representation U of E6 can induce the representation t4(w, q) of Y6on 9'. From Lemma 3.1,
-(f, Jsh),
(Jsf, h )
(Plcf, h?
=
-(f,P&),
=
1,293,
for all f , h E 9,where (. , .) is the inner product on S'.I n terms of the operators (6.27) these conditions become (JY, h? =
(J%
h ) = (.f, JW),
(PY,h )
=
< f , P3h?,
@+f, h )
=
(f,P-h).
(6.28)
T h e representation t4(w, q) is determined by Eqs. (6.5) where -2q is a 1, -q 2,...; m = -u, nonnegative integer, u = -q , -q --u 1,..., + u . If we assume the basis vectors are in 9 and apply the requirements (6.28), the following restrictions emerge:
+
+
(1)
w
(2)
( f z l
(3)
fz)
is real,
,f$")
=0
unless u = u' and m = m',
lz(u - m ) = ]:fI
lz(u
+ m + I),
+
258
6. THE EUCLIDEAN GROUP I N 3-SPACE
valid for all U E Qand m vectors {p:)}, u = -4, -q
= -u,
-u
+ 1,...; m
+ I , ..., +u.
= - u,
-u
Defining new basis for 9 by
+ 1,..., + u ;
we obtain 1 p:) I = j p:,‘) 1 for all m, m‘, u, u’ in the spectrum of the representation. (Th e phase factor (has been chosen for convenience.) Without loss of generality the vectors p:) can be assumed to be of length one. Hence, they form an orthonormal basis for X . Expressing relations (6.5) in terms of the basis p:) and setting s = -q, we find: Theorem 6.3 T h e possible faithful irreducible representations of 8,induced by an irreducible unitary representation U of E, on a Hilbert space X are determined by the pair ( w , s), where w is real and 2s is an
integer. T h e representation ( w , s) of 4 acts on % as follows: X has an I ,..., +u. orthonormal basis {p:)}; u = j s I, 1 s 1 +1,...; m = -u, -u I n terms of this basis the operators (6.27) are defined by
“
+
p”’:
J+pE) = [ ( u + m
+ l)(u - m)]l/zp:jl J-p;’ = [(u + m)(u - m + 1)]1’2p2!1 , u + + I)(u - m + l)(u + s + I)(u s + 1) = -w (u + 1)y2u + 1)(2u + 3) mws (u + m)(u m)(u + s)(u s) ] pp’ w [ J3p2)= mp:) ,
u(u
-
[(u
-
r)z
+ 1)
,
-
-
+ m + I)(u
uy2u
-
m)]1/2
ws
u(u
+ 1)(2u - 1)
+ 1) *m+1
\U)
-
1~
p2-1) ,
6-4. THE EUCLIDEAN G R O U P
259
E6
T h e representations ( w , s) and ( - w , -s) are isomorphic; otherwise all of the representations are distinct. (Note: T h e fact that the representation ( w , s) is defined for s negative follows from an examination of Eqs. (6.30). Even though the representation ( w , s) for s < 0 is isomorphic to ( - w , -s) it is convenient to consider ( w , s) in its own right.) Theorem 6.3 lists the possible representations of 8, induced by unitary irreducible representations of E 6 . We will now prove that each of these Lie algebra representations ( w , s) actually does originate from some unitary irreducible representation (also called ( w , s)) of E, . This will be done by constructing the unitary representation. According to expression (6.25) the finite group elements of EB are $ , , k = 1 , 2 , 3. uniquely determined by the infinitesimal generators Pk, Therefore, the unitary representation ( w , s) of E, on a Hilbert space .% is uniquely determined by the infinitesimal operators P, , Jk , k = 1 ,2 , 3, on 3 ; hence, by Eqs. (6.30). Indeed if U(r, A ) , (r,A ) E E, , is a unitary operator on .% corresponding to ( w , s) then
T h e construction of the irreducible unitary representations ( w , s) of E, is well known (Wightman [l]), and the results can be presented in an elegant coordinate free manner. Here, however, the construction will be carried out in a heavily coordinate-dependent manner to show the explicit connection between E, and the recursion relations (6.13) for special functions. From the orthogonality relation (5.144) for the matrix elements U,nm(A)on the group manifold SU(2) it folows that for fixed s the functions
u = is
1,
Isi
+ 1,...;
m
= -u,
-u
+ 1,..., +u;
form an orthonormal set with respect to the measure dT y 77, 0 a < 277, (mod 277);
0
< <
<
=
sin y dy da,
Furthermore, from (6.1 I), (6.15), and (6.29), the functions p:)(y, a ) satisfy Eqs. (6.30) with
6. THE EUCLIDEAN GROUP IN 3-SPACE
260
+
Here z = cos y , t = -ieia, and P:*"(cos y ) = e(-in/2)(s+m) B~~"(cos y i0). From these results it seems reasonable to define a Hilbert space 3? of functions square integrable with respect to dr such that the functions p$) given above form an orthonormal basis for Z . T h e differential operators (6.32) can then be used to determine a multiplier representation of E, on Z which should turn out to be the desired unitary representation ( w , s). This procedure works but it leads to a complication. For s an integer the p:)(y, a ) are functions of e i w , while if s is not an integer the basis vectors are functions of eia12. Because of this difference in behavior the two cases have to be treated separately. T o avoid these complications we adopt the device of multiplying each of the functions p g ) by eisa. Thus, we define new functions h k ) ( y ,a ) = eisor$k)(y, a)
+
Since m s is always an integer, whether or not s is an integer, this device allows us to treat both cases at once. It is obvious that the set (h:)} is still orthonormal with respect to the inner product defined above: (h$) , h'"')
=
1,j
257
hk;)(y,a ) h z ) ( y ,a ) dT = 6,.,m~u.,,.
0
Furthermore, from the results in Section 4-8concerning transformations of differential operators under mappings of function spaces, it follows that the basis functions h:) satisfy Eqs. (6.30) where now the differential operators are given by J3 =
-i-
a acu
J*
- s,
P*
=
a
eiia (& -
aY
= we*ia sin y ,
+ zcot y -aa + s
P3
= w cosy.
Moreover, the infinitesimal operators P, , J k become
a
J~ = sin a - + cos a cot y 2Y
P,
= -iw
sin a sin y ,
a
-
aa
+ is cos a ('1 siny cosy) '
P,
= -iw cosy.
THE EUCLIDEAN GROUP
6-4.
261
€6
These operators will be used to construct the unitary irreducible representation ( w , s) of E6 on X ( w , s), where X ( w , s) is the Hilbert space of all functions ~ ( C Oy ,S&a) square integrable with respect to the measure d7. T h e inner product is given by ( f,g> =
1
a
2a
0
0
.f (cos y , cia) g(cos y , eie) sin y dy da,
From the remarks immediately following Eq. ( 5 . la),concerning the completeness of the matrix elements U&(A) on the group manifold SU(2),we can conclude that the functions h z ' ( y , a ) , u = 1 s 1, 1 s I +l,...; m = - u , -u 1,..., + u ; form an orthonormal basis for X(u,s). I t will now be shown that the operators U(r, A), defined by (6.31) with infinitesimal generators (6.33), form a unitary irreducible representation of E, on %(w, s). T h e construction of these operators is a straightforward application of Theorem 1.10:
+
[ u ( r , ~ ) j l ( c oys, ei.1
=
exp(--iwr *
*
2)
f (cos y', cia'),
[a(cos yy ++ 1)1)++6bsinsinyy eia1 ~(COS
e-iu
fE
(6.34)
s),
#(W,
where r
3
*
j; =
C
,
f = (xl, x 2 , x3) = (cos a sin y , sin a sin y ,
y.x..
3=1
cos y' = cos y =
+ (abeiu + 66e-ia)
sin y ,
ic2 a2
ei"'
(I - 2b6)
sin y eiw - 6 2 sin y e-ia - 2a6 cos y sin y ecia - b2 sin y eia - 26b cos y
A
=
(--ba-
cos y ) ; (6.35)
(6.36)
b) E SU(2).
6
+
Here, all bracketed quantities are of absolute value 1. T h e action of the operators U(r,-A)can be written in a more transparent form by identifying the elements of X ( w , s) with functions square integrable with respect to Lebesgue measure on the unit 2-sphere. Thus, we make the identification ~ ( C O Sy , &) = f(2) where the unit vector rZ is given by (6.35). Integration with respect to the measure d7 can now be interpreted as integration over the surface of the 2-sphere. In terms of this notation the action of the operators U(r, A ) on X(u,S) is
262
6. THE EUCLIDEAN GROUP I N 3-SPACE
where the orthogonal matrix R ( A ) is defined by (6.23). I n this form it is obvious that these operators are defined and unitary for every (r,A ) E E6 (the orthogonal transformation R(A-I) preserves the scalar product while the multiplier has absolute value 1). T h e group property
+ R(A)r',AA')
U(r, A ) U(r', A ' ) = U(r
is true by construction, but can also be verified directly. T h u s we have defined a unitary representation of E6 on X ( w , s) which induces the Lie algebra representation ( w , s) of F6 classified in Theorem 6.3. T h e group representation can be shown to be irreducible since its induced Lie algebra representation is irreducible. Furthermore, the group representations ( w , s) and ( - W , -s) are unitary equivalent since their induced Lie algebra representations are isomorphic. We shall not give direct proofs of these facts since the proofs are somewhat tedious. See, however, Naimark [2], Chapter 3.
6-5
The Matrix Elements of ( w , s )
We turn now to the problem of determining the matrix elements of the , operators U(r, A ) with respect to the orthonormal basis {hk'} of X ( Ws). For fixed u the functions I%',"'($), m = -u, - u 1 , ..., f u , form a canonical basis for the irreducible representation D, of SU(2). T h u s we have
+
c U;m(A)hp)(ji) U
[U(O,A)h2'](iZ)=
k=-U
where the matrix elements U,",(A) are given in terms of Jacobi polynomials by (5.143). Clearly the matrix elements of the operator U(0, A ) are (hlz"',U(0, A)hg'j = UZm(A)S V & . T h e computation of matrix elements of the operator U(r, I ) , I the identity element of SU(2),is somewhat more difficult. Before performing the computation it is necessary to introduce some supplementary material from Section 5-16. If A E SU(2) has coordinates (pl , 0, p2) we can write the matrix eIements U,",(A) in the form Lzm(Fl , 0, pJ = e - i n a ~ Tm~(#)e-inPz ~ 11,
where TZ,,(B)
=
(i)-
[i" ++ u
").! ( u (u
72)'
~
~
. ) ! ] 1 ' 2
m)!
P,n*m(COS
0).
6-5. THE MATRIX ELEMENTS OF (w,s)
263
T h e functions T ~ , m ( 6 satisfy ) the relations
(6.38) *
C ( U , m; u’, m‘ I I , m
+ m’)TL+nf,m+ml(6)r
where the C( * ) are Clebsch-Gordan coefficients for SU(2), (5.53). T h e basis elements h k ) ( y ,a ) = Qy,m(y) ei(m+s)-and the T functions are related by the simple formula
Furthermore, the spherical harmonics Yy(6,9)) are given by YY(0,F)
=
(- l)zQo,n(t))eim~.
By definition the matrix elements of U(r,I) are
J”:
(hjl“), U ( r , 1)h.g))= [v,n I w , s 1 u, m](r) =
J”
*
271
Q;,n(y)ei(n+x exp(--iwr
0
- ji)Q;,(y)ei(m+s)a
sin y d y da.
(6.39)
Writing r in spherical coordinates r
=
( r sin 6, cos y r , r sin 0,. sin y , , r cos Or),
we make use of the well-known formula m
~ X P ( ~* r) P
=
4~
C
1-0
c i ” j ( P ) y f ( e r, F r ) Y t ( 6 , , 1
FBI
k=-l
(6.40)
where the j l are spherical Bessel functions, (A.24) (Kurgunoglu [I], Chapter 4). Substituting this identity into (6.39), applying (6.38), and simplifying, we obtain [a, n I w , s I u, ml(r) =
. i”i(w.)YY-n(O,, *
C(Z, m
-
q r )C(1,O;v , s 1 u, s)
n ; v, n 1 u, m).
(6.41)
6. THE EUCLIDEAN GROUP I N 3-SPACE
264
T h e fact that the U operators form a unitary representation of E, allows us to derive a number of relations satisfied by the functions (6.41). Thus, unitarity implies [w,
72
I w , s I u , ml(-r)
I w , s I v,n](r).
= [u, m
Furthermore, the group property U(rl , I) U(r2, I) = U(rl leads to [v,n
I w, s I u, mI(r1 + r?,)
m
=
=
+ r 2 ,I)
h
1 1 [v,n I w9 s I h, qI(r1) h=IsI Q=-h
*
When s = v = n spherical waves
(6.42)
[h,4
~
w, s
I u, mI(r2).
(6.43)
0, this identity reduces to the addition theorem for
.jh(wrl)jdwrz) Y V , , vr,)Yz””(&, eiTJ . C(Z, 0;h, 0 I u , O)C(Z, m - q; h, q I u, m)
(6.44)
+
which was first derived by Friedman and Russek [I]. Here, r = r1 r2 . T h e group property U(r, A ) = U(0, A ) U(A-lr, I) = U(r, I) U(0, A ) implies
U;,~(A)[V, ,’=-u
n’ I
w, s
I u, m l ( ~ - l r ) =
[n,n
I w , s I u, m ’ ] ( r ) ~ ; , ~ ( ~ ) . (6.45)
+ 1 component quantity
Fix v and consider the 2v XAy;:(r)
rn‘=--u
I
= ([v,n 1 w , s u, m](r)),
n
= -8,
-n
+ 1,..., Sv, (6.46)
for some u, m. Define the action W of E, on X(r) by (in matrix notation)
T h e identity (6.45) shows that X(r)transforms like a spinor field of weight v ; in fact, under the action of SU(2) it transforms like the eigenvector h z ) of the irreducible representation D, . Furthermore, (6.43) shows
[Wa, z)X::;;l(r)
m
=
h
1 1 Ih, q I
h=lsl q=-h
w , s I u, ~ l W x g $ ( r ) .
(6.48)
6-5. THE MATRIX ELEMENTS OF (w,s)
265
Let M(w, s, v) be the complex vector space generated by all finite linear combinations of the v-spinor functions x:?;:; u = I s I, I s I + 1, ...; m = - u , -u + 1,..., +u. We can uniquely define an inner product (., .)* on M(w, s, v), linear in the second argument, conjugate linear in the first, by requiring
for all admissible u, u', m,m'. Completing M(w, s, v) with respect to this inner product we obtain an abstract Hilbert space %(w, s, v). Denote again by W the action of E6 on H ( w , s, v) induced by (6.47). Then, W i s a unitary irreducible representation of E, on H(w,s, v), unitary equivalent to the representation ( w , s). I n fact the unitary equivalence maps the basis vector h:) into the v-spinor ~2;:. Clearly, we can construct a representation of E6 unitary equivalent to ( w , s) for each 1,... . value of v = 1 s 1, 1 s 1 Fix a again and observe that the action of E6 on %(w, s, v) induces an irreducible representation of cY6 isomorphic to ( w , s). A tedious computation using (6.47) shows that the infinitesimal operators corresponding to this representation are
+
eiiq
a + i cot 0 a) a'p
fa0
+ S*,
53
= --i-
a a'p
+ s3, (6.49)
~3
a
=icose--z-ar
.sin 0
r
a
80'
.
where r = ( r sin 8 cos p, r sin 8 sin q ~ ,r cos 0) and
+
S*[% n I w , s ! u , ml = [(v f .)(v T n 1)1""[., S 3 [ V , . 1 w,s I u , m ] = n [ v , n I w , s I u , m ] .
i 1 I w , s ! u, 4, (6.50)
xz;:
By replacing the eigenvectors p g ) with the v-spinors and substituting expressions (6.49) into Eqs. (6.30) the reader can easily obtain a series of recursion relations for the matrix elements [v, n 1 w , s 1 u, m](r). Note also that the last of Eqs. (6.30) yields the identity (V2
+ w 2 ) [ vn, I
I
w , s u, mNr) =
0,
(6.51)
where V2 is the Laplacian. Thus the functions x&z(r) are spinor-valued solutions of the wave equation.
266
6. THE EUCLIDEAN GROUP IN 3-SPACE
T h e v-spinor functions defined above are of special importance in mathematical physics. It is known that a v-spinor Y(r) satisfying the wave equation (V2 w z ) Y(r) = 0 can be expanded as a countable linear combination of the spinors X&%(r) where m = -u, -u 1,..., + u ; u = 1 s 1, 1 s 1 + 1,...; s = -v , -v 1,..., +v. Such an expansion is useful because of the simple transformation properties of the x functions under the action of E, . Thus, our results yield recursion relations and addition theorems for spinor-valued solutions of the wave equation. T h e spinors x also have simple completeness and orthogonality properties. For these see Miller [2]. This concludes our analysis of the representation theory of E, . We have seen that the matrix elements of the translation subgroup of E, are the so-called spherical wave solutions of the wave equation. T h e matrix elements have been chosen with respect to a basis indexed by the irreducible representations of SU(2). By computing the matrix elements with respect to a basis indexed by the irreducible representations of the subgroup {(r, I ) } or {(r, exp .A3)}(isomorphic to a 2-sheeted covering group of E3), we could derive identities relating spherical wave solutions of plane wave or cylindrical wave solutions of the wave equation. This computation is not difficult but will not be attempted here. T h e representations ( w , s) ( w real, 2s an integer, ( w , s) ( - w , -s)) derived above constitute all faithful irreducible unitary representations of E, . T h e only remaining unitary irreducible representations of E, not unitary equivalent to a representation ( w , s) are those which map the translation subgroup {(r,I ) } into the identity operator. Thus, on the subgroup SU(2) they must be unitary equivalent to a representation D, of SU(2). For proofs of these statements see Wightman [l] and Mackey [l].
+
+
+
CHAPTER
7 The Factorization Method
As presented in this book the procedure for associating special functions with a Lie algebra 9 divides naturally into three parts: (1) Determine an abstract irreducible representation p of 9.(2) Find a realization of p acting on a space lZ? of analytic functions. ( 3 ) Compute the multiplier representation of the local Lie group G induced by p on a. T h e practical importance of this procedure is demonstrated by numerous examples in Chapters 3-6. I n the next two chapters, however, we shall discontinue the multiplication of examples and concentrate instead on an analysis of the procedure itself. T he functions of hypergeometric type (hypergeometric, confluent hypergeometric, and Bessel functions) are solutions of linear second order ordinary differential equations and satisfy differential recurrence relations. E, F I n fact, corresponding to each of the operator types A, B, C‘, C”,D’, we have been able to derive a set of recurrence relations for these functions. Conversely, in this chapter, it will be shown that under rather general conditions the only “reasonable” families of functions satisfying second order linear differential equations and differential recurrence relations are the functions of hypergeometric type listed above. Furthermore, all of the recurrence relations obeyed by such functions can be obtained as combinations of recurrence relations derived earlier in this book. (There are families of functions other than functions of hypergeometric type, which satisfy second order equations and differential recursion relations, but they are too complicated to be of much practical importance.) In other words special functions associated with the Lie algebras %(a, b ) and ,T6completely exhaust the supply of functions with the
7. THE FACTORIZATION METHOD
268
simple properties listed above. T h e material of Chapters 3-6 describes a one-to-one relationship between the representation theory of these Lie algebras and recurrence formulas of functions of hypergeometric type. I n this sense Chapters 3-6 form a complete unit. T h e technique for proving the above results is the factorization method, originated by Schroedinger and due in its definitive form to Infeld and Hull [I]. This technique was developed to solve eigenvalue problems appearing in quantum theory but is also a very powerful tool for studying recurrence formulas obeyed by special functions. T h e version presented here is a slight modification of Infeld and Hull [l]. For more details, especially concerning applications to physics, the reader should consult this reference.
7-1
Recurrence Relations
+
Let X, = -d2/dx2 D,(x) dldx + Em(%) be a sequence of second order ordinary differential operators defined for m E S = {m,, m, f 1, m, f 2, ...}, where m, is an arbitrary complex number; and analytic for x in a neighborhood N of xo E @. We shall be interested in solving the eigenvalue problems X,Y,(m,
x) = AYA(?T2, x)
(7.1)
for all m E S and x E N . (In the following the dependence of the function Y,(m) on x will be supressed.) If there exist linear differential operators
defined for all m
E
S, x E N , such that
where theamareconstants, we say the operators X,admit a factorization. I n this case the eigenvalue equation (7.1) is equivalent to the two equations
for all m E S. T h e significance of the existence of a factorization for the solution of the eigenvalue problem is given by:
7-1. RECURRENCE RELATIONS
269
Lemma 7.1 Let Y,(Z) be a solution of (7.1) for m = E. Then Lr+lYA(l)is a solution of (7.1) for m = 1 1. Similarly L;Y,(Z) is a solution for m = 1 - 1.
+
PROOF
T h e second assertion is proved in exactly the same way.
Q.E.D.
Thus the operators L* satisfy the properties: (1) the “raising operators” L‘t,, map a solution of (7.1) for m = 1 into a solution for m = 1 + 1; (2) the “lowering operators” L t map a solution for m = E into a solution for m = 1 - 1 ; and (3) if we first raise and then lower (or vice versa) we obtain our original function multiplied by a fixed constant. (The possibil= 0 or L;Y,(Z) = 0 is not excluded.) Conversely, it ity that Lt+lYA(E) is not difficult to show that, by suitable renormalization, any set of linear differential operators Lg , m E S, satisfying properties (I), (2), and (3) can be assumed to satisfy (7.2) for some choice of constants a, . If the operators X , admit a factorization, the raising and lowering operators can be used to derive recurrence relations for the functions Y,(m). For example, if Y,(E) is a solution of (7.1) for m = I, a ladder of solutions YA(l n) can be defined recursively by
+
Then Eqs. (7.2) imply Y,(Z
+ n)
=
(A
-
u,+,,,)-”~L,,+,Y*(z
+ n + I),
n
= 0, 1 ,
2, ... .
These expressions are differential recurrence relations for the functions Y,(1 n), n 3 0. Similarly we can derive recurrence relations for - n) with n 3 0: functions YA(E
+
From this point of view the existence of differential recurrence relations satisfying properties ( I ) , (2), and ( 3 ) listed above is equivalent to the existence of a factorization. Moreover, all of the differential recurrence
270
7. THE FACTORIZATION METHOD
relations derived for special functions in Chapters 3-6 can be obtained as factorizations of operators X,, by raising and lowering operators L& . I n fact, all of the known recurrence relations for functions of hypergeometric type listed in the Bateman Project (Erdtlyi et al. [l]) can be derived either as factorizations or as combinations of recurrence relations obtained from factorizations. I t is thus a problem of considerable importance in special function theory to classify all possible factorizations. At first glance this classification problem appears very difficult-we must find all second order operators X,, , first order operators L& , L& , and constants a, for all m E S such that Eqs. (7.2) hold identically. However, the problem can be drastically simplified by transforming the operators X , into the standard form
To obtain these operators choose a family of analytic functions { q m ( x ) } such that y,(x) f 0 for all m E S , x E N , and define the mappings CpL1:
Y,(m, x) + Yi(m,x) = (pm(x))-lYA(m, x ) ,
T h e domain of '9 ;
mE
s.
is the space M(A, m) of all solutions of the equation X,Y,(m, x )
= AY,(m, x ) .
Denote the range of q;' by &'(A, m).Then the elements of &'(A, eigenfunctions of the second order differential operator
x;,= 9L1'(x) &v,(x),
m) are
[P)L1(x) = (Pm(4)-lI.
I n fact, X;Y;(m, x )
=
9)G1(x) Xmv,(x) P)G1(x) Y h , x ) = pL1(x)X,Y,(7%
= ApLl(x)
Y&, x)
4
= AY;(m, x).
Similarly the first order operators
M(A,m ) +&(A,
+ l),
m
m - 1)
L,:
A ( A , m)+&(A,
L,':
d"(A, m)+M'(A, m
are transformed into the operators L;l;l:
A ' ( A , m )+ A ' ( A ,
rn
+ I),
where
L2L1 = rp,:l(x)L;+lql,(x),
L-' WI
= P)rn-l(")Lrnrpm(X). --I
-
1)
7-1. RECURRENCE RELATIONS
271
As a consequence of these definitions the transformed operators L z form a factorization of the operators X g : L-’ L+‘ ,n+l
+ amtl = LZL;’ + a,
=X&,
m
E
S.
+
If we choose ~ , ( x ) = exp Jz0 D,(t) dt for all m E S , the transformed operators Xk take the form (774) in some neighborhood of xo E (2.’ In fact, X k = -d2/dx2 E,,(x) D k ( x ) - dD,(x)/dx. [This is a standard computation (see Petrovsky [l], Chapter 2).] Thus, without loss of generality it can be assumed that the differential operators are
x, =
+
+
+
d2 dx2
+ Vm(x),
L;
+ a,+,
= LLL-,
--
=
d
A;@) &
+Bi(x),
(7.5)
where L,+lL;+l
+ a,
= x,
(7.6)
for all m E S. Necessary and sufficient conditions for the validity of (7.6) are (1) AAA,
=
-1,
+ A,BA + B,A+, = 0, ALA,’ + A,BL + B,AL = 0, BL+I’L+I + ’G+IBZ+~+ = 3ABZ + AAB;;,’ + am
(2) A,AL’
(3) (4)
am+,
(7.7) 1
vm(x)
for all m E S , where now the prime denotes differentiation with respect to
x. Differentiation of (1) yields AZA; = -A&A;’. Comparing this expression with conditions (2) and (3) we obtain the requirements
Thus, A& and A; are constants such that AAA; = -1. Since (7.6) imposes restrictions only on products of the operators L,+,, L; , there is no loss of generality involved if the operator L , is assumed to be normalized so that A; = 1 for all m E S. From this assumption there follow the conditions
+
and
7. THE FACTORIZATION METHOD
272
T h e last two equations imply the equality (7.10)
where the constants a, are independent of x. Formula (7.10) is obviously a necessary condition for the existence of a factorization. However, it is also sufficient since functions B,(x) and constants a, satisfying (7.10) can be used to unambiguously define the functions V,(x) by (7.9), hence, to define operators X , which can be factored. Thus to find all possible factorization types it is sufficient to find all possible solutions of (7.10). 7-2
The Factorization Types
As a first trial solution of (7.10) choose B,(x) = f ( m ) where f is an arbitrary function of m. Then (7.10) is satisfied for a, = -f(rn)2 and (7.1) becomes the equation of simple harmonic motion
_ - d2 Y,(m) = hY,(m). dx2
(7.11)
This factorization is of such a trivial nature that we will not examine it in any detail. T h e trial solution B,(x) = j(x) mk(x) yields more useful results. Substituting this solution into (7.10) and equating powers of rn on both sides of the resulting expression we obtain the conditions
+
k2 + k’ = -a2,
j’
+ kj = 4 4 2 ,
am = a2m2
+ bpm,
(7.12)
where a, b, p are constants. (Since (7.10) depends only on the difference a, - a, , if the constants a, satisfy (7. lo), so do the constants a, c. However, it is easy to show that without loss of generality we can set c = 0. Indeed, c can be absorbed in the constant A, Eq. (7.1).) For a # 0 the possible solutions of (7.12) are:
+
Type ( B )
k(x)
= a cot a(x
k(x)
=
ia,
+p ) ,
j(x) =
~
1 x+p ’
= ___
2a
ibp 2a
where p and q are constants. If a Type (C‘) k(x)
bCL cot a(x j(x) = -
+ qe-iacs+P),
=
+ p ) + sin a(x4 + p ) ’
0, b # 0, the possible solutions are:
j ( x ) = - bP(X
+PI
4
~
4
.
x f p ’
7-2.THE FACTORIZATION TYPES Finally, if a
=
k(x) Type (D”)k(x)
b
=
=
0 the solutions are: 1
~
273
=
x+p’
= 0,
j(x)
=
. X 4f p ?
q.
Clearly, these factorizations correspond exactly to the type A-D” operators classified in Chapter 2. Indeed Eqs. (7.12) and (2.30) are identical. T h e type A and B factorizations can be normalized so that a 1, b = 0 by replacing a(x + p ) with x f p and m with m‘ = m - bp/2a2. In this case the expressions for k(x), j(x) corresponding to type A and B factorizations coincide with those for the type A and B operators. Similarly the expressions for k ( x ) , j ( x ) corresponding to type C’ and D’ factorizations coincide with those for type C’ and D’ operators when a = 0, b = 1. If a = b = 0 the expressions for k ( x ) , j ( x ) corresponding to type C” and D “ factorizations and type C“ and D“ operators coincide. Each of the factorization types listed above determines a family of second order ordinary differential equations 1
V,(x)
=(j =(
+ ( m + 1)k)2 +j’ + ( m + 1) k’ + az(m + + @(m + 1)
j + mk)2 -j’
-
mk
+ a2m2 + bpm,
which admit a factorization by the linear differential operators L+ m
1
-
d -+ j dx
+ mk,
L , = z +dj + m k .
(7.14)
T h e differential equations and recurrence relations for special functions given by these factorizations are almost exactly the same as those obtained from realizations of irreducible representations of the Lie algebras 9 ( a , b) by the corresponding operator types. T h e difference is merely one of normalization. For example if a = 1, b = 0, the type A factorizations are associated with the differential equations
*
YA(m,x)
= hY,(m, x).
(7.15)
274
7. THE FACTORIZATION METHOD
Set m' = m + +, h' = h - 2, q' = -q, and map the space A ( h , m ) into the new space of solutions Y i f ( m ,x) = q~;'(x) Y,(m, x) where q~;l(x) = (sin(x + p ) ) l i 2 (see Section 7-1). We find that the new solutions satisfy the transformed equations -
1 [sin(x sin(x + p ) dx
d
mt2
+ qI2
+ p ) dxd Y;,(m,x -
- 2q'm'cos(x sin2(x p )
+
)I
+ p ) ] y i , ( m , x) = A ' Y ; , ( ~x),
which are identical with the equations for eigenfunctions of type A operators derived in Section 2-7. Also, the recursion relations obtained from factorization by the operators LA , L; transform into recursion relations for type A eigenfunctions obtained from the operators Jf, J-. Similar remarks hold for the other factorization types. For more details concerning the derivation of identities for special functions from the factorization types A-D" the reader should consult Infeld and Hull [l]. Clearly, the theory of these factorization types is completely equivalent to the representation theory of the Lie algebras 9 ( a , b). Hence, the principal results of Infeld and Hull related to such factorizations are already contained in Chapters 3-5. (Infeld and Hull combine the type C' and type C" factorizations to form type C factorizations. Similarly, they combine types D' and D"to form type D factorizations. However, from the Lie algebraic viewpoint adopted in this book, each of the type C and type D factorizations should be broken up into subclasses; one subclass corresponding to faithful representations of Y(0, 1) and the other to faithful representations of Y(0,O). T h e type D" factorizations are a special case of the trivial factorizations corresponding to Eq. (7.1 I).) T o find additional solutions of (7.10) one might introduce the trial function
+ 1 k , ( x ) mz, n
B,(x) =.j(x)
kn(x)
1=1
+ 0,
m E S,
where n 3 2. However, the only possible solution in this case is the trivial B,(x) = f ( m ) considered earlier (see Infeld and Hull [I], p. 28). One trial function which does lead to new results is B,(x)
=
~
h(x) rn
+j ( x ) + mk(x),
m E S.
7-2.
THE FACTORIZATION
275
TYPES
I n fact, substituting this expression into (7.10) and equating powers of m on both sides of the resulting formula we can obtain the following new solutions: Type (E) h(x) a,
Type(F)
=
=
q,
k ( x ) = a cot a(x
j(x) = 0,
+ p),
a2m2 - q2/m2;
h(x) = q,
j(x) = 0,
k(x)
=
l/(x + p ) ,
a, = --Q2/m2,
where a,$, q are constants. T h e differential equations admitting type E factorizations take the form ( a = 1,p = 0)
[-
d2
d ~+ 2
1 YA(rn,x)
drn + l) + 29 cot x sin2x
= hY,(rn, x).
T h e solutions of this equation are closely related to the type A eigenfunctions. Indeed, transforming the equation to a new normal form by means of the substitutions
i z we obtain d2
+
3
-
=
2iq cos x
X
In tan - , W
2
=
sin-lla x Y,(m),
+ h + $ ] W ( z )= -(m +
W(z).
sin2 x
I n terms of the constants r , I given by X = - I 2 expression becomes d2
2rl cos z
+ r2 + ( I + 3)( sin2 z
] w(z)
- !i)
=
-(rn
- r2, q =
+
&)2
irl this
W ( z ) (7.16)
which can be identified with the type A equation (7.15). Thus, the eigenfunctions corresponding to type A and type E factorizations can be identified. (They are hypergeometric functions.) However, the type A recurrence relations raise and lower the I-value of these eigenfunctions whereas the type E recurrence relations raise and lower the m-value. This is the same kind of behavior exhibited in Section 6-2. There the type E operators were used to find realizations of irreducible representations of Y6. T h e basis vectors for these realizations were hypergeometric functions and the representation theory yielded two kinds of differential recurrence relations. T h e first class of recurrence relations was associated with the representation theory of 4 2 ) and we have seen
276
7. THE FACTORIZATION METHOD
that it is equivalent to the type A factorization. However, the second class of differentia1 recurrence relations, (6.14), was not directly associated with a Lie algebra representation, but rather with a representation of the By means of an appropriate change universal enveloping algebra of F6. of variable it is not difficult to show that this latter class of recurrence formulas is equivalent to the formulas obtained from the type E factorizations. Similarly the recurrence relations obtained from type F factorizations are equivalent to Eqs. (6.19) derived from a representation of the by type F differential operators. Here universal enveloping algebra of F6 again the type B and type F factorizations yield different recurrence formulas for the same eigenfunctions. (In this case the confluent hypergeometric functions.) T h e eight factorization types A-F derived above constitute the complete list of factorization types given by Infeld and Hull. Indeed, it is a straightforward exercise to show that the key equation (7.10) has no solutions of the form
c n
Bm(4
=
l=-nl
&(4mZ,
where n, n’ are finite nonnegative integers, other than the factorization types already listed. We have seen that each of the factorization types A-F corresponds to one of the differential operator types A-F studied in Chapters 3-6. Th u s there is a complete equivalence between the factorization method as formulated by Infeld and Hull and the study of realizations of irreducible representations of th-e Lie algebras 9 ( a , b) and F6 by differential operators in two complex variables. For this reason we will not delve deeper into the theory of the factorization method. It should not be supposed, however, that the factorization types of Infeld and Hull constitute all possible factorizations. I n fact, if f(x) is an arbitrary analytic function and (a,], m E S = (m,, rn, 1, m, 2, ...I, is a set of arbitrary complex constants, there is a solution B J x ) of (7.10) such that B,u(x) = f ( x ) . Such a solution (not unique) can be constructed recursively from (7.10). Th u s there are an infinite number of factorizations in addition to those we have listed. However, for none of the new factorizations can the function B,(x) be expressed in the form (7.17), i.e., in terms of a finite number of powers of m. I n view of this these new factorizations have not proved to be of much practical importance. Furthermore such factorizations seem to have no reasonable Lie algebraic interpretation.
*
*
CHAPTER
8 Generulized Lie Derivatives
As stated earlier, the method we have used to associate special functions with a Lie algebra 9 consists of the three steps: (1) Determine an abstract representation p of 9.(2) Find a realization of p by generalized Lie derivatives acting on a space QI of analytic functions. (3) Compute the multiplier representation of the local Lie group G induced by p on a. T h e first of these steps constitutes a problem in the representation theory of Lie algebras, a subject which has undergone intensive investigation (Jacobson [l]). T h e third step can be handled by the use of Theorem 1.10. Thus, for low-dimensional Lie algebras at least, steps (1) and (3) can be carried through on the basis of existing results in the mathematical literature. By constrast, step (2) is not a well-defined procedure treated in the literature. For the purposes of special function theory we need to classify “all” realizations of 9 by generalized Lie derivatives. That such a classification is important follows from our study of the Lie algebras 9 ( a , b) and . We found realizations of these Lie algebras in terms of generalized Lie derivatives in two complex variables, operator types A-F. This choice of operators completely determined the possible special functions which could arise as basis vectors. There is no a priori guarantee, however, that there are not other realizations of these Lie algebras by generalized Lie derivatives in two variables which lead to new classes of special functions. (In fact it will be shown there do exist realizations of 9 ( 0 , 1) and 9(0,0 ) by generalized Lie derivatives distinct from the ones we have studied, but that they lead to no new classes of special functions.)
8. GENERALIZED LIE DERIVATIVES
278
T o solve such problems we will develop the rudiments of a classification theory of generalized Lie derivatives. Significant results in the classification theory of ordinary Lie derivatives are well known (Lie [l], Vol. 111). I n particular, Lie himself found all possible realizations of complex Lie algebras by Lie derivatives in one and two complex variables. For more than two variables the results are incomplete. Here we shall extend Lie's methods slightly so as to apply to generalized Lie derivatives. Among the results which will be obtained from this analysis are a proof by that the operator types A-F are the only realizations of 9 ( a , b) and F6 generalized Lie derivatives in two complex variables which lead to interesting special functions, and a table listing all Lie algebras which have a realization in terms of generalized Lie derivatives in one complex variable. I n Chapter 9 the table will be used to derive new classes of special functions which are not of hypergeometric type. T h e material of this chapter is more difficult than that of the rest of the book. I n particular, fairly deep results from the theory of local Lie groups and the cohomology theory of Lie algebras will be employed. Therefore, those readers approaching the subject for the first time may prefer to omit Chapter 8 on a first reading.
8-1
Generalized Derivations
Here, notation and terminology for generalized Lie derivatives and multiplier representations will largely follow that which was introduced in Section 1-3. However, it will prove convenient to introduce an alternative definition of the generalized Lie derivative which has the advantage of being independent of local coordinates. Given a point xo E @%, let O? be the space of all functions analytic in some neighborhood of xo,i.e., the germs of functions at xo (Gunning and Rossi [l], Chapter 2). (Without loss of generality we can assume xo = 0 = (0,..., O).) O? is a complex vector space, but it is also an associative algebra, since the product of two functions in 67! is a function in 67!. T o expose the algebraic structure of the results in this chapter we introduce the concept of a realization of 9 by generalized derivations operating on an abstract associative algebra A. When A = this concept is identical to the notion of a realization of 9 by generalized Lie derivatives on
a
a.
Definition Let A be an associative algebra over @with a multiplicative identity 1. T h e mapping D of A into A is said to be a generalized derivation (gd) on A if
279
8-1. GENERALIZED DERIVATIONS
af) + +
+
(1) D(.f bh) = bD(h), (2) W f h ) = fD(h) D ( f ) h - D(1)fh for all a, b E Q and f,h E A. If in addition D(1) = 0, where 0 is the additive identity of A, then D is an (ordinary) derivation on A. If D is a gd on A , it is easy to verify that the operator D = D - D(1) A , is a derivation on A. Thus, defined by D( f) = D(f) -- D ( l ) f , f ~ D = D D(1) and D is uniquely defined by this relation. Conversely, h, defined by if D is a derivation on A and h E A , the operator D = D D( f) = B( f) h f , f ~A , is a gd on A. I n fact h = D(1).Note: We are using the same notation for the operator D(1) and the element D(I ) in A. This should cause no confusion. If D,, D, are gd’s on A the commutator [Dl, D,]= D D - D,D, .z is a gd on A. Indeed, a straightforward computation gives
+
+
+
- w,
[Ol, 4-J = - D,D, and [Dl, D,1(1) = m D , ( l ) > - DZPl(1)). For a, b E Q the operator aD, + bD, has an obvious definition as a gd on A . Thus, the gd’s on A form a (possibly infinite-dimensional) Lie algebra. I n terms of the local coordinates x = (xl, x, ,..., x,) in a neighborhood of 0 E a gd D on the algebra 0 2 can be expressed uniquely in the form
for allfE Ol (see Cohn [I], p. 17). Here,
( T he multiplicative identity of Ol is the constant function l(x) = I for all x E Q,.) Thus, on the algebra a, the definition of generalized derivations is equivalent to the definition of generalized Lie derivatives given in Section 1-3. Definition A realization
A is a mapping (1)
Tna+bp
(2)
%,B1
01
+T,
+ bT3
= = CT.
where
7
T
T,
of a complex Lie algebra 3 by gd’s on is a gd on A, such that
7
TO1
for every a, b E @, 01, /3 E 22. Th e T is transitive ( A # a) if ?*(f)= 0, for all 01 E 22 implies f = a l , a E Q ; or ( A = if [?,(f)](O) = 0, all 01 E implies [p(f)](O) = 0 for all derivations p of a.
a)
8. GENERALIZED LIE DERIVATIVES
280
I n the case A = 02, this definition is equivalent to the coordinatedependent definition of a realization of 9 given in Section 1-3. Moreover, the following theorem holds. Theorem 8.1 Let 3 be a complex Lie algebra and G the local Lie group such that '3 = L(G). There is a one-to-one correspondence between realizations T of 9 by gd's on and local multiplier representations T.of G on 02. T h e relationship between the multiplier v ( x , g ) and the gd's T , is given by
T
is transitive if and only if T.is transitive.
PROOF (Rough sketch) If T is an effective realization of 9 (or if T.is effective) this theorem is merely a reformulation of Theorems 1.9 and 1.10. Suppose then the set JV = {a E 9:T , f = 0 for all f E is a
nonzero ideal of '3. We must show there exists exactly one multiplier representation TY of G whose infinitesimal representation is T . Uniqueness follows from Theorem 1.9. To prove existence note that T induces an effective realization T' of the quotient algebra 9iN. (The elements of '3/Ncan be denoted as a' for all a E 9.[We will sometimes write a' = a M . ] Then a' = ,Y if and only if a - ,t3 E N .Addition of elements and multiplication by a scalar are defined by aa' ZIP' = (aa ZIP)', a, b E G', a, p E 9.T h e commutator (., .) of 31.1. is given by ( a ' , p') = [a, PI' where [., .] is the commutator of 9. These operations on are well defined since N is an ideal of 9.) Here, T:, = T , for all a E 9. According to Theorem 1.10, T' induces an effective multiplier representation TY' of the local Lie quotient group GIN where N is a normal subgroup of G. [Here 9 = L(G),JV = L ( N ) , ~ / J V =L(G/N) (see Pontrjagin [I], Chapter 9).] Moreover, the canonical projection 40: G - t GIN is a local analytic homomorphism. Thus the operators TY(g) on 02 defined by T"(d = ' w d g ) ) > g E G ,
+
+
+
determine a local multiplier representation T.of G whose infinitesimal Q.E.D. representation is T . Lemma 8.1
Let
T
Then, (1) (2)
[Fa , "I
"(Ta(l>>
be a realization of 9 by gd's on the algebra A.
= %J31,
- ?a(Te(l)> = Tra,aI(l>,
8-2. COHOMOLOGY CLASSES OF REALIZATIONS
+
281
b?, , nTe(1) bTa(1) for all a , b E G, a, ,6 E 9.I n particular the Fa form a realization b of by derivations on A. Conversely, derivations b, and elements T,( 1) of A defined for each a E 9 and satisfying relations (1)-(4) determine a T,( 1). realization T of 9 by gd’s on A, where T , = ?,
(3)
?,,+bb
(4)
T,,+bS(1)
= a.7, =
+
+
PROOF If T is a realization of 9 every gd T , can be decomposed T,( 1). If we carry out this decomposition uniquely in the form T , = 7, on both sides of the identities
+
= aTcx f b T f l (1)’ (2)’ %,a1 = [ T . 7al we see that (1)’ and (2)’ are equivalent to conditions (1)-(4). Q.E.D. 9
9
8-2
Cohomology Classes of Realizations
TO be a realization of 9 by derivations on A, i.e., ~ , “ ( 1 ) = 0 for all 9.Denote by A ( T O ) the set of all realizations T of 9 by gd’s on A such that b = T O . T h e A ( T O ) can be given the structure of a vector space over @: If T , T’ E A(T”)and a , b E @, define the realization @ b ~ E’ A(T”)by
Let
aE
(UT
+b~;(l),
@ b ~ ’ )(1) , =UT~(I)
all a E 3.
T h e additive identity element in the vector space A(T”)is T O . Clearly, to classify all possible realizations of 9 by gd’s on Gt it is enough to find all of the vector spaces @ ( T O ) where T O runs through a complete list of possible realizations of 9 by ordinary Lie derivatives (derivations). T h e problem of classifying the realizations of by derivations has been studied in detail by Lie and completely solved for Lie derivatives in one or two complex variables ( n = 1, 2). For larger values of n the results are incomplete (Lie [l], Vol. 111). Here, we shall fall back on Lie’s results and assume we know all of the realizations T O of 9 by derivations on (This is equivalent to the knowledge of all possible ways in which G can act as a local transformation group on an n-dimensional complex manifold.) T o complete the classification of all possible realizations of 9 by gd’s we need only compute the vector space a(T O ) for each T O . This is still a difficult problem since in general a(,.) is infinite-dimensional. T o simplify the computations involved in the
a.
282
8. GENERALIZED LIE DERIVATIVES
solution of this problem note that, for the purposes of special function can be conveniently divided into theory, the elements of @(TO) equivalence classes. This can be seen as follows: Let p E @ where ~ ( 0# ) 0. Then, 9-l E @. This function defines vector space isomor) and phisms cp and cp-l of GZ onto GZ such that cp[f](x) = ~ ( xf(x) cp-l[f](x) = p-l(x) f(x) for all f E a. Similarly p induces an isomorphism cp' of the vector space TO) onto itself. Here,
+ ~ 4 1 +) F-~+,(F)
~'[71, = 'P-~T,'P = ?,
(8.3)
for all T E @(TO), E 9 (see Lemma 4.4).T h e function p(x) can always /k! be expressed in the form p(x) = exp(f(x)) = C z = o ( f ( ~ ) ) k where f E O! ( f i s not unique). Then (8.3) becomes (Y
VP"T1, = ?,
+
+ Tdf).
(8.4)
I n Chapters 4 and 5 it was shown by means of examples that the special function theory associated with the operators 7, is completely equivalent to the special function theory associated with the operators cp'[T], . Clearly, the special functions corresponding to the two sets of operators are related by the multiplicative factor p(x). Using this discussion as motivation we make the general definition: Definition Two realizations 7, 7' of 9 by gd's on A are said to be cohomologous, 7 GZ T ' , if there is an f~ A such that ?, = T : , 7:(1) = ~ ~ ( 1 ~) , ( f ) ,all E 9. By definition c'w''is an equivalence relation on A ( 7 O ) . Thus, A(T")is divided into equivalence classes of cohomologous operators. Moreover, this equivalence relation respects the vector space structure of A(T").T h e =) { p E A(T"): p GZ TO), i.e., the equivalence class containing T O set N ( T ~ is clearly a subspace of A(T").Furthermore, it is easy to show T w T' for T , T' E A(T")if and only if 7 = 7' @ p for some p E N(7O). This proves the existence of an isomorphism between the set of equivalence classes I n particular the set of in A(T")and the quotient space A(T~)/N(TO). equivalence classes can be assigned a vector space structure. As suggested by the term "cohomologous," the above result can be formulated in terms of the cohomology theory of Lie algebras. We follow Jacobson's presentation [l, Chapter 31. Let 7" be a realization of 9 by derivations on the algebra A. If i 3 1, an i-dimensional A-cochain for 9 is an i-linear mapping f which associates to each i-tuple (oil ,..., m i ) , ai E 9, an element f ( a l ,..., mi) of A in such a way that f is skew-symmetric in its i arguments. A O-dimensional A-cochain is a map + h where E 9 and h is a fixed element of A. T h e collection
+
(Y
(Y
(Y
8-2. COHOMOLOGY CLASSES OF REALIZATIONS
283
C i ( 9 , A ) of all i-cochains for A forms a complex vector space under the usual definitions of addition and scalar multiplication of functions. T h e coboundary operator 6 is a linear mapping of C i ( 9 , A ) into Ci+l(9,A ) ,i >, 0, defined by
-f([m2
3
a317 011).
(8.7)
An i-cochain f is a cocycle if Sf = 0 and a coboundary iff = 6h for some h E Ci-'. Clearly, the set Zi(Y, A ) of all i-dimensional cocycles and the set B"9, A ) of all i-dimensional coboundaries are subspaces of Ci. From the definition (8.5) it is a standard argument to show that a2 = 0, i.e. Bi_C Zi.T h e factor space H i ( 9 , A ) = Zi(9,A)'Bi(9,A ) is called the i-dimensional cohomology space of $2 relative to A. I n the above discussion we have defined the cohomology spaces of 9 relative to an algebra A. However, the same construction goes through for a module, A,and one can define cohomology spaces H i ( 9 , A) (See Jacobson [l], Chapter 3 ) . We shall have occasion to make use of this remark later. A comparison of our study of A(T")and the cohomology theory for the pair 9,A is illuminating. HO(9, A ) can be identified with the space of all elements h E A such that 7 3 = 0, all a E g.Thus, 7" is transitive implies that dim HO(9, A ) = 1. From Lemma 8.1, T E A(+) if and only if ~ ~ ( is1 a) 1-cocycle. I n fact, this correspondence establishes an isomorphism between the vector space 4 7 " ) and the space Z l ( 9 , A ) of 1-cocycles. T w o realizations T , T' E A ( T O ) are cohomologous if and only if their corresponding 1-cocycles differ by a coboundary, i.e., T w T' if and only if T : ( I ) = ~ ~ ( + 1 )8f, f E Co(Y, A). T h e space N(7") can be A). identified with the space of I-coboundaries B1(9,
8. GENERALIZED LIE DERIVATIVES
284
Theorem 8.2 A(TO)/N(TO)= W ( 3 ,A). T h e W(3,A ) is isomorphic to the space of equivalence classes of realizations of 3 by generalized derivations on A. We return now to the special case A = a. Suppose TO is analytic in an open set U containing 0 E Cn. According to Theorem 8.1 there is a one-to-one correspondence between elements of 0 1 1 ( 7 O ) and elements of the set X ( T ~consisting ) of all multiplier representations TY of G on such that the action x -+ xg, of G as a local transformation group on U is induced by 9. T h e operators P ( g ) are given by
where Y is a multiplier (see Section 1-3). Put another way, there is one-toand complex functions Y one correspondence between elements of on U x G such that
a(+)
for x in a suitably small neighborhood of 0 and g, , g, in a suitably small neighborhood of the identity element e of G. I n particular the multiplier vo(x,g)= 1 corresponds to TO. We can use this one-to-one correspondence to induce a vector space structure and an equivalence relation on X(+'). T h u s if the multiplier representations T.,TY' are associated with the realizations 7, T ' , respectively, we require the sum TYY' to be the multiplier representation associated with the realization 7 @ 7' and the scalar multiple T."of TY by a E e( to be the multiplier representation associated with a7. Similarly we say TY is cohomologous to TY' (TY- TY') if and only if T rn 7 ' . T h e result of this construction is the definition: Definition T h e sum of TY, P'E X(T")is the multiplier representation P'E X ( F ) where vv'(x,g) = Y(X, g) v'(x,g); and the scalar multiple of T. by a 6 e( is the representation T " E X ( + ) where va(x,g) = [ ~ ( xg),]". T h e representations TY and TY' are cohomologous ( T. P') if T.'= q-lTYq for some q , i.e., if ~ ' ( xg), = v(x,g) v(xg)/v(x) for some v E ~ ( 0# ) 0. According to these definitions X(7O) is a vector space isomorphic to @(TO). Moreover the factor space X(7")/X0(+')is isomorphic to H1(g,a) where X o ( ~ 0= ) (TY E X ( 9 ) : TY P o } is the cohomology class of the identity element of X(70). These relationships can be used to derive more information about H 1 ( 3 ,a).
-
a,
N
8-2. COHOMOLOGY CLASSES OF REALIZATIONS
285
At this point we make the assumption that T O is transitive or, what is the same thing, that the elements of X ( T ~are ) locally transitive on U. Intransitive multiplier representations do not seem to be of much interest for the purposes of special function theory. In particular, all of the multiplier representations studied in this book and applied to the functions of hypergeometric type are locally transitive. Let K, be the isotropy subgroup of G with respect to the point X E U , i.e., K , = (g E G: xg = x}. It is well known that the groups K , , K, corresponding to any two points x,y in a sufficiently small neighborhood of O E U are conjugate subgroups of G (Pontrjagin [l], Chapter 9). There will be no loss of generality if we restrict ourselves to consideration of the isotropy subgroup K = K O .[Since G is locally transitive and K is closed in G the set U can be identified with an open set in the local homogeneous space GIK and the action of G on U as a local transformation group can be identified with the natural action of G on G / K by right multiplication (Pontrjagin [l], Chapter 9)]. Let TY E X ( T ~ )Since . G is locally transitive it is possible to choose an element g, E G for all x in a suitably small neighborhood of 0 such that Og, = x and the function x +g, is analytic in the coordinates of x. From (8.9) there follows the relation
v(0,klk2)
=
v ( 0 , kl) 4 0 , h),
k, I k , , klk2 E K *
Thus, p(k) = v(0, k) defines a local l-dimensional representation of K. Furthermore, v(0,kg)
= ~ ( 0k,) ~ ( 0g), ,
k E K , g, kg E G .
This equation and (8.9) yield the useful relation (8.10)
where k(x,g) E Kis defined byg,g = k ( x , g)g,, . If we set ~ ( x = ) ~ ( 0g,) ,, then y E p(0) # 0, and (8.10) becomes
a,
(8.11)
Conversely, if p E GZ, ~ ( 0 # ) 0, and p is a local 1-dimensional representation of K , by reversing the above argument it is easy to show that the function v defined by (8.11) satisfies conditions (8.9), hence, defines a multiplier representation TY of G.
8. GENERALIZED LIE DERIVATIVES
286
Comparing (8. I 1) with the definition of cohomologous multiplier representations we find TY TY‘ if and only if p(k) = p’(k) for all k E K . Thus to determine all noncohomologous multiplier representations of G with given action on U , we need only determine the possible analytic 1-dimensional representations of K . If p is an analytic 1-dimensional representation of K , then p maps all commutators in K into 1 E (2’: N
tL(klk2k;lk;l)
= P(k1) d k 2 ) P@l)-l
CL(kJ1
=
1.
Let K* be the commutator subgroup of K , i.e., the closed normal subgroup generated by the commutators k,k,k,lk;l of K . I n this case p ( K * ) = 1 and p induces a representation p* of the abelian factor group Kl K * : k
p*(kK*) = p ( k ) ,
E
(8.12)
K.
Suppose K / K * has dimension s as a local Lie group. Then there are local coordinates (xl ,..., xs) for KIK* in a neighborhood of 0 E gssuch that the coordinates of eK* are (0,..., 0), and if k,K*, k,K* have coordinates (al ,..., us), (b, ,..., bs), respectively, then (k,K*)(K,K*)= k,K* has coordinates ( a , b, ,..., a, b,). Since p* is a l-dimensional analytic representation of an s-dimensional abelian group, there must exist complex constants c1 ,..., c, such that
+
p * ( k K * ) = p*(xl ,..., x,)
+
=
exp(x,c,
+ x2cp + ... + x,c,)
(8.13)
where (x, ,..., xs) are the local coordinates of kK* E KIK*. Conversely, if complex constants c1 ,..., c, are given and p* is defined on K / K * by (8.13), we can define a representation p of K by (8.12) and thus determine a multiplier representation T of G. There is a one-to-one correspondence between classes of cohomologous multiplier representations of G and complex s-tuples (cl ,..., cs). Indeed, two multiplier representations are cohomologous if and only if they are associated with the same complex s-tuple. T h e set of all complex s-tuples has a natural vector space structure which agrees with the induced structure of the space of cohomology classes. I n fact, if TY, TY’ E X(+)are associated with (cl ,..., cs), (c; ,..., ci), respectively, then an easy computation shows P ’ is associated with the s-tuple (cl c; ,..., c, 6:). Similarly, TYa is associated with the s-tuple (acl ,..., uc,). As a consequence of the above construction there follow the equalities
+
+
dim Hl(9, 02)= dim X ( T O ) / X O ( T O )
=
dim K
-
dim K*
= s.
8-2. COHOMOLOGY CLASSES OF REALIZATIONS
287
These equalities prove that the cohomology space HI(%, a)is finitedimensional and that each cohomology class is associated with a unique s-tuple (cl ,..., cs). Such results suggest the possibility of a practical method for computing the cohomology classes. We can use the isomorphism between X(r') and a(+') to phrase our conclusions in terms of differential operators. Theorem 8.3 Let ro be a transitive realization of 9 by derivations on 6Y and let X be the isotropy subalgebra of B with respect to TO,
Then, if X * = [ X ,-XI is the derived algebra of X , we have dim H O ( 9 , a ) = 1, dim H 1 ( 9 ,
a)= d i m s
-
dimX*.
PROOF A'- = L ( K ) ,X * = L(K*). Here X * is the Lie algebra generated by the set of all commutators ,B], U , ,B E X .Theorem 8.3 is of fundamental importance for the computation of cohomology classes for realizations of 9.I n fact, given a transitive realization r of 9 on CE we can read off the dimensions of X and X * directly and thus determine the dimension of H 1 ( 9 ,0') We . will use these results in Section 8-3 to explicitly compute realizations r in each of the cohomology classes of a(+'). Before leaving the abstract theory, however, we make several observations: [(t,
t
(1) Let T be a realization of B by gd's on 0% and let 3 be the subalgebra of 9 defined by 3 = {a E 9:r, f = 0, all f E 0%>.I t is easy to show that 9is an ideal in 9,[9, $1 C 9.Thus, T induces a realization T' of the factor algebra 8' = 9/9. ( T : + , ~ = T, for all 01 E 9.) Now T' is an effective realization of 9'in the sense that if 01 Y E B', 01 4 3,then T:+$ # 0. As a consequence of this fact there is no loss in generality if we study only those realizations of 9 which are effective. Thus, in the remainder of this chapter only the spaces of realizations of B by gd's such that at least one 7 E @(TO) is effective will be considered. (2) Let T E a(.")be an effective realization of B and consider the set A? = {(t E 9:TE f = 0, all f E a}. Clearly, A 2' is an abelian ideal in 9:
+
a(+')
288
8. GENERALIZED LIE DERIVATIVES
If dim A?”
= m, the set of functions ~ ~ (E1GZ) defined for all 01 E A form an rn-dimensional vector space @(A) in @, (If dim @(A) <m there exists an 01 E A such that T, = 0, in which case T is not effective.) a(&)can be given the structure of an abelian Lie algebra by defining the commutator of any two elements to be O E ~ ( & ) . Then the Lie and ? @(A) are isomorphic and can be identified. 9 acts on algebras &i A by the derivations TO,
induces an effective realization TO’ of the factor algebra 9’ = 9 j A on A obtained in an obvious manner from (8.14).Thus, given an effective realization of 9 by gd’s on GZ we have constructed an effective realization of the factor algebra 9’ by derivations on a(&). Conversely, let po be an effective realization of a Lie algebra 9‘ by derivations on an m-dimensional vector space G Z ( 4 ) C GZ. We impose upon GZ(d2‘) the structure of an abelian Lie algebra and identify it with the abstract m-dimensional Lie algebra A.By reversing the argument in the preceding paragraph and using the theory of extensions of Lie algebras (Seminaire Sophus Lie [l], Cartan and Eilenberg [l]), we can find a Lie algebra 9 containing 4as an ideal and an effective realization 9‘ (neither 9 nor T is unique) such that 914 T of 9 by gd’s on and T induces the realization p o of 9‘ on 4.I n fact, it can be shown that the possible extensions 9 of A by 9‘ are in one-to-one correspondence with the elements of the space H2(9’,a(&))where the action of 9’on a(&’) is given by po. I n the case where 9‘ is semisimple we can apply the Whitehead lemmas (Jacobson [l], Chapter 3) which state H5(9’,a(&))= 0, i = 0, 1, 2, ... . Thus, H*(9‘, a(&?)) consists of the zero element alone and there is only one possible extension of A by 9‘.As is well known, that extension is 9 = 9‘@ A,the split extension of 9’by A (Jacobson [l], Chapter 1; Seminaire Sophus Lie [l]). TO
Theorem 8.4 Let 9 be a Lie algebra and T a transitive effective Let 4 = (E E 9:?- E 01, 9’ = 914 as realization of 9 by gd’s on 9‘ @ A. defined above. If 9’ is semisimple then 9 T h e above results constitute a “bootstrap” method for constructing realizations T of a Lie algebra 3 by gd’s from a knowledge of the realizations po of lower-dimensional Lie algebras 9‘ by derivations. This method will prove useful in the next section. Note that the Whitehead lemmas are not applicable to the computation of the cohomology spaces is infinite-dimensional. H i ( 9 , GZ), i = 0, 1, 2,..., because
a.
8-3. COMPUTATION OF THE COHOMOLOGY CLASSES
289
8-3 Computation of the Cohomology Classes T h e abstract theory presented in the last section will now be applied to explicitly construct the possible realizations of a Lie algebra. Let T O be a transitive realization of 9 by derivations on a with isotropy subalgebra
x,
T
= {m E
9: [ T i f ] ( o ) = 0,
allf E
a}.
= dim 2 9 we can choose a basis a1 ,..., a,. for 9 and define the structure constants C f j with respect to this basis by
If r
(8.15) 1=1
As is well known (Jacobson [l], Chapter l), the skew-symmetry of the commutator and the Jacobi equality imply the relations
c!.= Cf. a)
+ c;c:j + Cj”lC,”J
T
9 1 ,
C
(CX,”,
p=1
<
<
for 1 i, j , I, b r. According to (8.15) the Lie derivatives j = 1,..., r, satisfy the relations T
YjYk
-
c7kYD,
YkYj =
(8.16)
=0
Yi defined by Y j = T:, , 1 <j, k
< r.
(8.17)
p=l
I n terms of local coordinates in U C @n the form
Yican be expressed in the
a. a(+‘)
where Pj E Since T O is transitive the r x n matrix (Pj(0))has rank n. If T E the functions fjE defined by fj = ~ , j ( l ) 1, j ,< r, satisfy the relations
<
c 7
Y,(fk)
-
Yk(fj)
=
p=1
< <
c:kfp,
1 <j, k
< r.
(8.18)
Conversely, any set { f , } , 1 j r, of elements of that satisfies (8.18) uniquely defines a member T of @ ( T O ) . I n fact T,, = Yi fj. Thus, the problem of determining all members of a(+)is the same as the problem of determining all solutions fi ,...,fT of (8.18).
+
8. GENERALIZED LIE DERIVATIVES
290 If
T , T'
E
GZ(T"), then
T
M
T'
if and only if
f: =f, + Y ? ( f ) ,
1<j
< y,
(8.19)
for some f E 02, where fj = T J 1), f j = T ' , (1). T o find a realization in of each cohomology class of G Z ( T O ) we need 'to find all solutions (8.18) and then from among these solutions pick out one element in each equivalence class (8.19). @(TO) is in general infinite-dimensional so this seems to be a difficult problem. However, use of Theorem 8.3 simplifies the computation considerably. For simplicity assume there exists an n-dimensional subalgebra 2 of 3' such that 9 = 2 X (direct sum as a vector space), where X is the isotropy subalgebra of 9 with respect to T O . I t is not necessary that A? be an ideal in 9.If this condition is satisfied, we say T O is regular. Clearly, if T O is regular the restriction of T O to the subalgebra A? yields a transitive realization of 2 on GZ with isotropy subalgebra zero. Regular realizations are of frequent occurrence. I n fact, an examination of Lie's tables (Lie [l], Vol. 111, pp. 4-5 and 71-73) listing the local transformation groups operating effectively on spaces of one and two complex variables shows that all of these realizations satisfy this condition. Furthermore, most of the realizations T O studied in this book also satisfy it. T o recapitulate, we are concerned with an r-dimensional complex Lie algebra 9 and a regular realization T O of 9 by derivations on the space GZ of analytic functions of n complex variables. 9 = Z X where A? is an n-dimensional subalgebra and X is the isotropy subalgebra of 9 with respect to T O . Let dim X = k , dim Z* = s where X * is the derived 0. We can choose a basis subalgebra of X . Clearly k = r - n ,..., 0 1 ~for 9 such that 0 1 ~,..., 01, form a basis for 2 and ,..., ar form a basis for X . In this case the Lie derivatives Yl ,..., Y , are linearly independent in a neighborhood of 0 E 2f.' while [Yi f](O) = 0 for all f E 02 and j = n 1,..., Y. Since To/# has isotropy subalgebra zero we know from Theorem 8.3 that T/# m T " i 2 for any T E (This follows from the fact that W ( 2 ,GZ) = 0.) T h u s any element of @(TO) is cohomologous to a realization T for which f, = ~,,(1) = 0, 1 j n. Furthermore, if T , T' E 02(~") with ~ ~ ~=( ~1:?)( 1 )= 0, 1 j n, then T rn T' if and only if T , = T: for all cy. E Z. Applying Theorem 8.3 again we have:
{fi}
+
+
+
a(+').
< < < <
Lemma 8.2 If T O is regular, there is an isomorphism between the ( k - s)-dimensional space of cohomology classes of @ ( T O ) and the space of solutions (fi, f, ,...,f?) of Eqs. (8.18) such thatf, = f i = .*. =f,,= 0.
8-3. COMPUTATION OF THE COHOMOLOGY CLASSES
291
T h e most general solution of this kind depends linearly on k - s constants of integration and the choice of these constants determines the cohomology class of the solution. As an example we will find a realization in each cohomology class of the space a ( 7 - O ) where T O is the realization of 4 2 ) by differential operators
T h e realization
T
E Q!(T")
takes the form
where
-df2 _ dz
z-=ffi dfl dz
'
df3 - 2 2 1 -df 2 -
dz
dz
f2,
z -df - 3z 2 2 - df - f 3 .
dz
dz
(8.21)
Clearly, 3? is the 2-dimensional subalgebra of 4 2 ) with basis {a2,a3)and
X * has basis {a3).For 2 we can choose the subalgebra with basis (a1).
Then, Theorem 8.3 implies dim H1(sZ(2),@) = 2 - 1 = 1. According to Lemma 8.2, 7 is cohomologous to a realization for whichf,(z) = 0. Thus, Eqs. (8.21) reduce to df2 dz
- = 0,
2zf2
=f 3 ,
with solution Tal
d dz '
=-
T~~
=z
d
+ c, dz
T,$
d
= 22 -
dz
+
~CZ,
c E G'.
(8.22)
T h e value of the constant c determines the cohomology class in which 7 lies. All other solutions of (8.21) are cohomologous to a solution (8.22) and no two solutions of the form (8.22) are cohomologous unless they are identical. T h e general solution of (8.21) is
L
where f is an arbitrary element of
a.
8. GENERALIZED LIE DERIVATIVES
292
T h e above example was so elementary that the cohomology classes could easily have been computed without recourse to Theorem 8.3 and Lemma 8.2. However, for higher-dimensional Lie algebras these results assume considerable practical importance. Nonregular realizations of a Lie algebra do not obey Lemma 8.2 and it is much more difficult to compute explicitly the cohomology classes of such realizations. An example of a nonregular realization is (Section 8-4). I n Sections 2-7 and 2-8 we computed realizations of 4 2 ) that seem to violate the analyticity assumptions made in this chapter. In particular the differential operators
c2
Tq
d
=&
-;, P
T~~
d
=z-
dz
+ A,
T , ~ =
d
z2 dz
+ (2h + p) z
(8.23)
form a realization of sZ(2), where A, p are complex constants. If p # 0 this realization has a singularity at z = 0. Thus, T $ for p # 0. However, if T is transformed to the realization T' = 9 - l ~ where ~ q is the function q(x) = z p , then
a(.")
d2=
d
+ c,
c=A+p.
(8.24)
Clearly, T' is an element of a(.") in the normalized form (8.22). (Note: Equation (4.82) remains valid even if q(0)= 0 or q I n Chapter 5 it was shown that the realizations T for p # 0 could be used to construct certain infinite-dimensional irreducible representations of $42). T h e basis functions for these representations were hk(z)= zk,k = 0, 5 1 , 1 2 ,... . However, it is easy to show that these same representations could be ~ on the space generobtained from the realizations T' = C P - ~ Toperating ated by the functions hL(z) = q-'(z) hk(z)= zk-p, k = 0, & I , f 2 , ... . Thus, it was not necessary to use the singular operators in Chapter 5 ; we could have obtained the same results in special function theory by using the analytic operators T' and the basis {h;). T h e singular operators were used simply because the basis {hk}is more convenient for computational purposes. There is thus no loss of generality in restricting ourselves to elements of TO). These remarks hold true for all of the realizations of the Lie algebras 3 ( a , b) and F ,, by differential operators as computed in Chapters 3-6. I n each case where the realization T has a singularity at the point xo = 0 we can find a function q(x) such that the realization T' = ( P - ~ T ' is ~ an element of There is no loss of generality for special function theory involved in the restriction to elements of a(?'). Note, however, that in the construction of irreducible representations of 9 in terms of a reali-
a.)
@(?I).
8-4. TABLES OF COHOMOLOGY CLASSES
293
zation T we do not necessarily assume that the basis space for this representation is a subspace of a. T h e elements of the basis space may have a singularity at xQ = 0.
8-4 Tables of Cohomology Classes As a first application of our results we list all transitive effective realizations of Lie algebras by gd’s acting on functions of one complex variable, i.e., n = 1. T o determine this list it is sufficient to use the methods indicated in observation ( 2 ) ,Section 8-2. Lie has proved that, to within an analytic change of variable, the only transitive effective realizations of a Lie algebra by ordinary derivations in one complex variable are (Lie [l], Vol. 111, pp. 4-5)
(8.25)
I n each of the three possible cases the differential operators corresponding to a basis of the associated Lie algebra are given. I n particular, (3) is the realization of 4 2 ) studied in the last section. For n = 1 the only possibilities for the factor algebras 9 defined in observation ( 2 ) are the three Lie algebras determined by (8.25). Corresponding to each of these three cases we must find all finite-dimensional subspaces A of cpl which are invariant under the action of 9‘on cpl as given by (8.25). (Such a classification is most easily obtained by finding a basis for 4 such that the matrix of d/dx is in Jordan canonical form.) ’ an abelian Lie algebra we must Considering each such subspace ~ 2 as then determine u p to isomorphism all possible Lie algebras g3A‘ such that 3’/& 9’. T h e list of all Lie algebras 9 so obtained exhausts the list of Lie algebras which have a transitive effective realization in terms of gd’s in one complex variable. T h e above computation is straightforward but tedious so we shall merely list the results. For each Lie algebra in the list we give an effective realization T . All other realizations in the set a(?)can easily be obtained from T by using the methods presented in Section 8-2. For each realization the values of r = dim 9,k = dim Z, s = dim Z* will be given. Recall dim If1(%, a) = k - s.
8. GENERALIZED LIE DERIVATIVES
294
Theorem 8.5 If T’ is a transitive effective realization of a finitedimensional Lie algebra by gd’s in one complex variable, then either T’ E @(?) for some T in the following list or T’ can be obtained from such a realization by an analytic change of variable. T h e list is: d
dz ’
&.
a
l i = 0 , 1 , 2 ,..., m i ,
i = l , 2 ,..., q,
2.
t e z ,
@, al(al
ai E
ai # ai for
- 1) = 0 ,
i # j.
r=q+I+Cmi,
m i , q nonnegative integers
K=r-l,
s=O.
(8.26)
0.
(8.27)
i=l
d dz
-
2, k = l ,
’
S =
(8.29) d dz
d
z-
dz’
22
d dz ’
-
1;
r=4,
k=3,
s=l.
(8.30)
All of the realizations of the Lie algebras $(a, b) obtained in Section
2-8 are special cases of this theorem. Thus, the realizations of 9 ( 0 , 0 ) are obtained from (8.26) by setting p = 3, a, = 0, a2 = 1, a3 = - 1, m, = m 2 = m3 = 0: s =
(Introduce the new variable z’ from (8.28), where p = 1:
=
3.
e“.) T h e realizations of g(0, 1) follow
T h e realizations of %(I, 0) are obtained from (8.30):
8-4. TABLES OF COHOMOLOGY CLASSES
295
There are no realizations of 9 ( u , 6) for n = 1 other than those given here. This proves that the restrictive assumptions made in Section 2-8 were not necessary; indeed, the most general realization of 9 ( u , 6) is cohomologous to a realization derived in Section 2-8. (Note that there is no by gd’s in one complex variable.) realization of F6 T h e algebraic difficulties involved in computing all transitive realizations of Lie algebras by gd’s in two complex variables are great enough so as to make it impractical to write down such a list. However, it is not difficult to find all possible such realizations of a given Lie algebra 9.Lie has computed a list of all possible realizations of Lie algebras by ordinary derivations for the case n = 2 (Lie [l], Vol. 111, pp. 71-73). T o find all realizations of a given Lie algebra 3 by gd’s it is necessary to classify the possible abelian ideals & of 9.(The ideals & and &’ are ’ A”.) identified if there is an automorphism of 9 which maps ~ 4 onto For each such A, Lie’s tables can be used to determine the possible realizations of the factor algebra 3‘ 9 / M by ordinary derivations. Each of these realizations of 9‘can then be extended to give realizations of 9.According to observation (2) at the end of Section 8-2, every realization of 9 can be obtained in this way. I n Chapters 3-6 it was shown that the functions of hypergeometric type were associated with realizations of the Lie algebras 4 2 ) ’ 9 ( 0 , 1)’ F3, F6 by the differential operator types A-F. It was not clear why those particular realizations were singled out for attention or whether there were other realizations in terms of gd’s in two complex variables. We shall resolve this problem by listing all transitive effective realizations of these Lie algebras for n = 2. T h e results are:
r=3,
k=l,
s=O.
r=3,
k=l,
s=o.
Every transitive realization of 4 2 ) by gd’s in two complex variables is, to within a change of coordinates, either an element of ctl(pl) or an element of LZ(p2).T h e type A and type B operators correspond to a selection of an element in each cohomology class in ctl(pl) and ctl(p2), respectively.
296
8. GENERALIZED LIE DERIVATIVES
a
z,-,
Xlaz,,
53:
a
r=4,
k=2,
r=4,
k = 2 , s=O,
a
a
- -; az, ’ az,
az,
r=4,
a
a
-, e Z i z z , e + - aZ2
54:
s=O,
az,
k=2,
s=l,
1; r=4,
k = 2 , s=O.
Every transitive effective realization of 9(0, 1) by gd’s in two complex variables is, to within change of coordinates, an element of one of the i = 1 , ..., 4.T h e type D’ and C’ operators correspond to a spaces and a([& selection of an element in each cohomology class in respectively. Th e realizations in and @([J have not occurred in this book up to now. However, as the reader can verify, they do not lead to any new results in special function theory.
a(&),
a([,)
a
a
ax2
az,
&:z2--z1-
a
-
a
-;
’ az, ’ ax,
r=3,
k=l,
s=O,
r=3,
k=l,
s=O,
r=3,
k=l,
s=O.
Every transitive effective realization of F3 by gd’s in two variables is, for i = 1, 2, or 3 . to within a change of coordinates, an element of a(&) T h e type D” and C “ operators correspond to a selection of an element in each cohomology class in Q?(4,) and a(E2), respectively. Realizations in did not occur in Chapter 3 , but they lead to no new special functions.
8-4. TABLES OF COHOMOLOGY CLASSES
e z l cos z 2 ,
-sin x2 ,
-ezi+zz,
e%,
cos z, ;
r
=
6, k
= 4,
e-%+Zz;
r
=
6, k
=
297
s = 2,
4, s = 2.
Every transitive effective realization of by gd’s in two complex variables is, to within a change of coordinates, an element of @&) or fl(&J.T h e type E and F operators correspond to a selection of an element in each cohomology class of @(PI) and a&), respectively.
CHAPTER
9 Some Generulizutions
U p to this point all identities for special functions have been derived from a study of the representation theory of the Lie algebras Y(a, 6 ) and F6. However, the Lie theory approach to special functions is of much wider applicability than these examples may indicate. T o bolster this contention we use the results of Chapter 8 to discover some new Lie algebras which have realizations by generalized Lie derivatives in one and two complex variables. Corresponding to each of the new Lie algebras we will derive identities for special functions by relating these functions to the representation theory of the Lie algebra. T h e first five sections of this chapter are devoted to a brief examination of the Lie algebra X 5 .A study of X5 leads to new identities for the Hermite functions, the best known of which is Mehler’s theorem, (9.36). T h e special function theory of .X, presented here is by no means exhaustive, and the interested reader is invited to derive additional results. (Weisner [2] has obtained identities for the Hermite functions even more general than those derived here, by studying a 6-dimensional Lie algebra which contains X5as a subalgebra. However, his 6-dimensional algebra has no realizations by generalized Lie derivatives in one complex variable.) I n the last half of the chapter we introduce a family of 3-dimensional Lie algebras YP,*which forms a natural generalization of F3 91,1. T h e special functions associated with this family form a natural generalization of Bessel functions, and the identities obeyed by these functions are analogous to those derived for Bessel functions in Chapter 3 . T h e above examples are indicative of the use of Lie theory for the study of special functions. This theory can be employed both to obtain
9-1. THE LIE ALGEBRA .X5
299
properties of known special functions and to derive new functions which are interesting enough to deserve the label “special.” Additional examples of the applications of Lie theory to special functions are found in Vilenkin [2], [3], notably the special function theory of unitary representations of the orthogonal and Euclidean groups in n-space.
9-1
The Lie Algebra X5
X5is the 5-dimensional complex Lie algebra with basis and commutation relations
+&F*, [y3,91= 2 9 [f-,$+l = 8, [ f - ,91 = 2y+, [f+, 91 [&*, 81 = [ f 3 , 81 = [ 2 ,&] = 0. [23,2-*1
f*,
23,
&, 9
=
(9.1)
=0
Clearly, the 4-dimensional subalgebra of X5generated by f * , f 3 , d is isomorphic to 9 ( 0 , 1). T h e Lie algebra X, is of interest to us because it has realizations by differential operators in one complex variable. I n fact, for q = 2 in expression (8.28) one obtains the operators a1:
d dz’
-
z-
d
1, z, 9 ;
dz’
r = 5,
k
=
4,
s = 2,
which define an effective realization of X 5 .Every transitive effective realization of .X, by gd’s in one complex variable is an element of
QYQ.
-X, also has realizations in two complex variables. Using Lie’s tables (Lie [l], Vol. 111, pp. 71-73), and the methods of Chapter 8 one can show that every transitive effective realization of X5 by gd’s in two complex variables is an element of r71(pj), j = 1,..., 5, where
k=3,
r=5,
p2:
a
a
8%
az,
z2--z1-,
a
a
a
+4 ;
tz,, - , 1, 22,az, 8x2 az, r=5, k=3,
-
r=5,
s=l;
k=3,
s=l;
~ = 2 ;
3 00
9. SOME GENERALIZATIONS
9-2 The Lie Group K5
K5 is the 5-dimensional complex Lie group with elements
=g(q
+ e2-q‘, a + a’ + eTcb’,b + eTb’ + 2eZTcq‘,c + e-‘c‘, + 7
7’).
(9.3)
I n particular the identity element of K5 is g(0, O,O, 0, 0) and the inverse of g(q, a, b, c, T) is g ( - q r z T , -a
+ bc
-
2c2q, -be-7
+ 2cqecT, -ceT,
-7).
T h e associative law can be verified directly. This group has the 5 x 5 matrix realization ceT beer 2a - bc T g(q, a, b, c, T ) =
0 0
0 0
where now the group operation is matrix multiplication. I t is clear from (9.3) that the set of all group elements with q = 0 forms a subgroup of K5 isomorphic to G(0, 1) (compare with (4.11)). A simple computation using (9.3) or (9.4) shows that the Lie algebra of K5 is isomorphic to -X,. I n fact we can make the identification g ( q , a, b, c, 7) = e x p ( q 9 ) exp(a&) exp(b$+) exp(c$-) e x p ( ~ f ~ )(9.5)
x3,
where the elements 2*, &, B generate X5and satisfy the commutation relations (9.1). Equation (9.5) uniquely determines K5 as a local Lie group. Moreover, as a global group K5 is simply connected.
9-3. REPRESENTATIONS OF .X5
9-3
301
Representations of X5
Rather than try to give a complete analysis of the representation theory of .X, we restrict ourselves to irreducible representations which can be realized by elements of a(ci,)acting on spaces of analytic functions. For such representations it will be easy to compute the matrix elements. I n particular, we study the irreducible representations p of X, satisfying property A : p / 9 ( 0 , 1) ( p restricted to the subalgebra 9(0, 1)) is isomorphic to one of the irreducible representations R(w, m, , p) or To,p of 9(0,1). Such representations will yield additional information about special functions associated with 9 ( 0 ,I). Let p be an abstract irreducible representation of X5on a vector space V which satisfies the above requirement and let
p(Y*)
=
J*,
=
J3,
p ( 4 = E,
p(2) = Q
be operators on V . These operators clearly satisfy the commutation relations
[J3, 5’1 [J-, Q]
=
=
[J3, Ql
&J+, [J+, Q]
2J+,
=
[J-, J+l = E,
2Q,
=
[J*, El
0,
=
[J3, E]
=
[Q, El
= 0.
Theorem 9.1 Every irreducible representation p of X, satisfying property A is isomorphic to a representation in the following list: (1) R’(W m, , Y),
T h e spectrum of
uT.:
J3
W > P E @,
< Re m,
Y
f 0, w
+ m, not an integer.
is the set S = {m,
+ n: n an integer}.
0
(2)
, p E @,
w , m,
fL
< 1,
# 0.
T h e spectrum of J 3 is the set S = {-LO + n: n a nonnegative integer}. For each of the above classes the representation space V has a basis {fn,),m E S , such that JYm =
mfm
J’fm
9
Efm
= /+frn+l>
= pfm J-fm
Qfm
(m
1
= pfrn+z
+
(9-6)
Wlfm-1
for all m E S on the left-hand sides of these equations. All of the irreducible representations in classes ( 1 ) and (2) satisfy property A and no two of them are isomorphic.
9. SOME GENERALIZATIONS
302
T h e proof of this theorem is elementary and is left to the reader. We could of course find many other representations of X 5 ,but for purposes of illustration, the representations listed in Theorem 9. I will suffice.
9-4
The Representation K(w, m, , p)
We will find a realization of R’(w, m, , p ) given by some T‘ E @(a,) acting on a vector space Vi of analytic functions of z. According to the theory of Section 8-2, the space of cohomology classes of a ( E l ) is 2-dimensional. Moreover, every element of @(al) is cohomologous to a realization T’ of the form
T w o realizations of the form (9.7) are cohomologous if and only if they are identical. T o construct a realization of R’(w, m, , p ) in terms of the operators (9.7) let Yi be the space of all finite linear combinations of the functions &(z) = z m + w , defined for all m e S where S is the spectrum of the representation, and let c1 = - w , c2 = p. Then
for all m E S. Comparison of the3e expressions with (9.6) yields a realization of R’(w, m, , p ) in terms of linear differential operators. w is not an integer the functions fA(z)= zm+ware not Since m analytic and single-valued in a neighborhood of z = 0. This is very annoying for computational purposes. T o remedy the defect we map the space Yi onto the space “y; = cp-’(V;)where cp-l[ f ’ ] ( z ) = (p(z))-lf‘(z) E Yl , f ’ E V i , and p(z) = Then Yl has a basis of the form fm(z)= cp-“ fT3(,z)= zk where K = m - m, is an integer. cp induces a transformation T = ( p - 1 ~ ‘ mapping ~ a realization T‘ of .X5 by gd’s on Y ; into a realization T by gd’s on Yl ; see (8.3). Under this
+
Z~O+W.
9-4. THE REPRESENTATION R’(w, rn, , p)
transformation the operators (9.7) with c1 = into the operators
-w,
303
c2 = p are mapped
on Vl . T h e operators (9.9) and the basis functionsf,(z) = zk,m = m, + k, provide the desired realization of R‘(w, m,, p). (Note that with the exception of Q = pzz, these operators and basis functions are identical with those derived for a realization of the representation R ( w , m a ,p) of 3 ( 0 , 1) in Section 4-1.) Following our usual procedure we can extend this realization of R’(w, m a ,p) defined on Vl to a local multiplier representation of K5 on G l l , where GTl is the complex vector space of all functions of z analytic in some neighborhood of the point z = 1. Here Ul 3 “y; and Ul is invariant under the operators (9.9). T h e operators A(g), g = g(q, a , b, c , T) E K5 , defining the multiplier representation can be computed i n . a straightforward manner from expressions (9.5) and (9.9). T h e result is
we can show that every function h E yl C Ul has a unique Laurent expansion
which converges absolutely for all 1 z I > d, where 1 > dh 3 0. (Tl is invariant under A ( g ) , see Section 2-2.) Thus, the functions f,(z) = hk(z)= zk, m = m, + k, in Vl form an analytic basis for T h e matrix elements A , , ( g ) of the operators A ( g ) with respect to this basis are defined by
vl.
m
[A(g)htJ(z)
C
l=--m
A&)hi(Z),
k
= 0 , fl, f2,-,
(9.11)
9. SOME GENERALIZATIONS
304
Since the operators A(g) define a local multiplier representation of K5we obtain the addition theorem 5
Azdgigz)
=
C
j=-m
I, k
Azi(gi)Ajlc(gB),
= 0,
i l , III~,...,
(9.12)
valid for g , , g, in a sufficiently small neighborhood of the identity element. To be precise we should determine the exact domain of validity of (9.12), just as in Section 4-1. However, it-will soon be evident that both sides of this equation are entire analytic functions of the group parameters of g , and g, . Thus, by analytic continuation equation (9.12) holds without restriction for all g, ,g , E K5 . T h e matrix elements A , , ( g ) can be evaluated directly from the generating function (9.11). We examine some special cases: (1) g ( 0 , a, b, c, T) ( q = 0 ) . I n this case (9.11) is identical with the generating function (4.8). Thus the matrix elements are proportional to the Laguerre functions,
+
where p = m, w is not an integer. ( 2 ) g(q, 0, b, 0,O). T h e generating function becomes exp[p(qz2
-+ bz)]zk =
m
Atk(g)z’. z=-5
Comparing this expression with the generating function (4.77) for the Hermite polynomials we find
Here, the matrix elements are entire functions of p and q. (3) g(q, 0, 0, c, 0). T h e generating function is
where R;+L(pq, c )
=
A,,(g). Thus, (9.15)
9-4. THE REPRESENTATION R’(w, m, , p)
305
where j ranges over all integral values such that the summand makes sense. There is no simple expression for the functions RP,in terms of functions of hypergeometric type. Various identities relating these matrix elements can be obtained from the addition theorem (9.12). (We have already derived identities for the matrix elements (9.13) in Section 4-1.) Thus, the relation g(q, 0, b, 0, 0 ) g(q’, 0, b‘, 0, 0 ) = g(4
+ 4’, 0, b + b’, 0, 0 )
implies (after some simplification) the identity (4
+4’Y2 n!
b
+ b‘ m=O
(9.16)
T h e relation
implies
There are many other identities contained in (9.12), but the reader can derive them for himself. I n analogy with Section 4-6 we can construct a realization of R‘(w, m, , p) by gd’s on a space of two complex variables where the action of -X, on this space is given by a member of @(fl1),(9.2). (The elements of a(&) when restricted to 9(0, 1) are identical with the type D’ operators.) From the theory of Section 8-2 it is easy to show that the space of is 2-dimensional and that every element of cohomology classes of a(&) this space is cohomologous to a realization of the form
Two realizations of this form are cohomologous if and only if they are identical. Note: T h e selection of a representative in each cohomology class given by (9.18) differs on the subalgebra 9 ( 0 , 1) from that chosen in Section 4-6. T h e representative selected here has been chosen for convenience in the computations to follow.
3 06
9. SOME GENERALIZATIONS
T o construct a realization of R’(w, m, , p ) using the operators (9.18) we must find nonzero functionsf,(x, y ) = Z,(x) emv such that Eqs. (9.6) are valid for all m E S = {m,+ n: n an integer}. I n terms of the functions Z,(x) these relations become (i) (ii)
(&d - P)z
m
= PZm+&43
d
-dx Z,(x) = ( m
+
W)
(9.19)
Zn-l(x),
Just as in Section 4-6 we can assume w = 0, p = 1 without any loss of generality for special function theory. Furthermore, a simple computation shows (i), (ii), and (iii) are compatible if and only if c = -1. Thus our problem reduces to finding a realization of R’(0, m, , 1) in terms of the operators (9.18) where c = -1. I n this case Eqs. (9.19) have the following linearly independent solutions for all m E S : (1) Z,(x)
=
(2) Z,(x)
=
(-1)m-m
0
2-m/zH,(x/2/2)
=
(- l)m-mo exp(x2/4)D,(x),
+ 1 ) 2(m+l)/zH-,-1(42/2) = exp(x2/4)e--iw(m+1)/2T(m + 1) D-,-l(ix). exp(x2/2)e--i7i(m+1)/21‘(m
(9.20)
T h e H,(x) are Hermite functions defined in terms of parabolic cylinder functions by H,(x) = 2mIz exp x2/2D,(d/zx). When m is a positive integer, which is not the case here, H,(x) is a Hermite polynomial. T h e fact that expressions (9.20) satisfy the recursion relations (i) and (ii) follows easily from (4.66) and (4.67). Relation (iii) can be derived from (i) and (ii). These solutions are entire functions of x. Clearly, if the functions 2, , m E S , are given by either (1) or (2), then the functionsf,(x, y ) = Z,(x) emu form an analytic basis for a realization of the representation R’(0,m, , 1) of X, . As usual this representation of S, can be extended to a local multiplier representation of K , by operators T(g),g E K , , on the space 9of all functions analytic in a neighborhood of the point (xo, yo) = (0, 0). A straightforward computation using (9.5) and (9.18) yields
~ E F (9.21) ,
where t
=
ev.
9-4. THE REPRESENTATION R'(w, m, , p)
307
T h e matrix elements of the operators T(g) with respect to a basis fm(x, t ) = ZJx) tm satisfying (9.19) are given by the functions A,,(g), (9.11), where w = 0, p = 1. Thus we obtain the relations
Corresponding to the basis
for all integers k, this equation can be written in the form x2t29 - txb
-
bV/2
x
+ bt
-
ct-1
-
2ct q
For q = 0, (9.22) is identical with (4.70) so we will omit this case. If a = c = T = 0, q = -$, (9.22) reduces (after some simplification) to the identity
(9.23)
valid for all m E @ not an integer. (In the next section we will show that (9.23) is also valid for m a nonnegative integer.) If a = b = T = 0, q = - $, (9.22) simplifies to
m
where h:(c)
=
(-
d/2)-n I?;( -4, z/%)
has the generating function
9. SOME GENERALIZATIONS
308
Here m is not an integer. However, in the next section we will show that this equation remains true for m a nonnegative integer. Similarly, the second set of basis vectors (9.20) can be used to derive identities for the Hermite functions. For example the case a = c = T = 0, q = -4, yields the identity
a generalization of (9.16). By substituting appropriate choices for the group parameter into (9.22) the reader can derive additional identities obeyed by the Hermite functions.
9-5
The Representation
f’w.p
I n analogy with the procedure of the last section we look for a realization of the representation t’u,pgiven by some T E 6Y(El) acting on a vector space of analytic functions of z. A comparison of expressions (4.17) and (9.7) shows how to proceed. Namely, in (9.7) we set c1 = - w , c2 = p to obtain J3
=
d
--
dz
w,
d
J- = dz,
J+ = pz,
E
= p,
Q
= pz2.
(9.24)
Designate by V2the space of all finite linear combinations of the functions zk, k = 0, 1, 2, ..., and define the basis vectors fm of V2by fm(x) = h k ( 2 ) = zk where m = -w k, k 3 0. Clearly, hk(z) =
+
JY, = (--w J-f,
+ z dzd l
zk =
mj,,
J+L
= (pz) z k =
+
,
Efm = /+fin ,
d dz
= - zk = ( m
Qfm
=
-w)fm-l
(P’)zk = d m + z
c~fn+~,
(9.25)
-
‘These relations define a realization of Tb,, on Vz. Moreover, this realization can easily be extended to a local multiplier representation B of K5on the space 6Y2 consisting of entire functions of z. T h e operators B(g),g = g(p, a, b, c, T) E K 5 , defining this representation are easily computed: [ B ( g ) f ] ( z= ) exp[p(p2
+ bz + a )
-
-w7]f(eTz
+ eTc), f~ rY2.
(9.26)
309
9-5. THE REPRESENTATION
Furthermore it is straightforward to show that the operators B(g) form a global representation of K 5 .T h e matrix elements B,,(g) of B(g) with respect to the basis (f, = hk) are given by m
LB(g) h k l ( Z )
=
Blk(g) hl(z), l=O
= 0,
1, &***,
or
T h e addition theorem for the matrix elements is (9.28)
valid for all g, ,g, E K5 . Corresponding to some special choices of the group parameters the matrix elements have the following explicit expressions: (1) g(0, a b, c, T). For q = 0, (9.27) becomes identical with the generating function (4.22). The matrix elements are thus proportional to associated Laguerre polynomials.
(2) g(q, 0, b, 0, 0). Exactly as in (9.14) we find
(3) g(q, 0, 0, c, 0). In this case (9.27) reduces to (9.30)
where R;-,(pq, c)
=
Blk(g).Thus, (9.31)
31 0
9. SOME GENERALIZATIONS
where j ranges over all integral values such that the summand is defined. Comparing (9.30) with the generating function (4.77) for the Hermite polynomials we find (9.32)
Just as in the last section, the addition theorem (9.28) can be used to derive identities obeyed by the matrix elements. This task will be left to the reader. we now Armed with a knowledge of the matrix elements of procede to construct a realization of this representation by gd’s on a space of two complex variables, where the action of X5is given by a member of a(&), (9.2). I n particular we will use the operators (9.18) with c = -p. T o construct a realization of using these operators it is necessary to find nonzero functions f,(x,y) = Z,(x) emu, m = - w , -a 1,..., such that Eqs. (9.6) are valid with
+
Just as in the derivation of Eqs. (9.19), it is easy to show that without loss of generality for special function theory we can assume w = 0, p = 1. Then, expressed in terms of the functions Zm(x),Eqs. (9.6) become the recursion relations
(9.33)
where m takes the values 0, 1, 2, ... . The functions Z,(x) are determined to within an arbitrary constant by (9.33). I n fact relation (ii) implies Z,(x) = c for some constant c. If we set c = 1, the remaining solutions Z,(x) are uniquely determined: Z,(x)
=
(-1)m
2-m/2Hm(x/z/Z)
=
(-1)m
exp(x2/4) Dm(x),
(9.34)
9-5. T H E REPRESENTATION
f'u,p
311
where the H,(x) are Hermite polynomials. Thus the vectors f , ( x , y ) = Z,(x) emy, m = 0, 1, 2 ,..., where Z,(x) is given by (9.34), form a basis for a realization of the representation of Z5. This infinitesimal representation of .X5can be extended to a local multiplier representation of K5 by operators T(g),g E K 5 ,on the space 9of all functions analytic in a neighborhood of the point (9, y o ) = (0,o). Indeed we have already computed the operators T(g) on 9, Eq. (9.21). T h e matrix elements of T(g) with respect to an analytic basis (fm(x, y ) ) satisfying (9.33) are the functions B,,(g) given by (9.27) ( w = 0, p = 1). Consequently,
c BZk(g)fZ(x,y), ' m
[ T ( g ) f k l ( x , y== )
=
1=0
O, 2 , * * * ~
or
(9.35)
0 this expression is identical with (4.76). For a
= c = T = 0, it reduces to (9.23) where now m is a nonnegative integer. I n the special case n = 0, (9.23) becomes
For q
q
=
=
-B,
(9.36)
which is Mehler's theorem, (4.158). Finally, if a = b = T = 0, q = -*, (9.35) reduces to the identity
W
=
C h$-,(c) H,(x) tZ-'-,
(9.37)
1=0
where hF-E--k(c)= (- 2/Z)k-zRf-k(-4, 4 Z c ) is given by the generating function m
exp(--z2/4)(z
-
2c)k
=
1 hl'_k(c)
21.
1=0
31 2
9. SOME GENERALIZATIONS
This completes our analysis of the special function theory of X 5 . Note that the identities for Hermite functions derived here have been obtained from a study of the operator realizations a1 and P I , (8.28) and (9.2). We could also have used p5 , but this would not have led to any new results. T h e operators p 2 , p 3 , p4, on the other hand, do not yield but rather lead realizations of the representations R’(w, m, , p ) or T;+, to realizations of representations of X5which we have not classified here. I n particular, p2 can be used to realize certain reducible representations in such a way that the basis vectors turn out to be proportional t o generalized Laguerre functions. Th u s a study of p2 leads to new identities for Laguerre functions.
9-6 The Lie Algebra 5??p,q Corresponding to a pair of positive integers ( p , q) let gPrq be the f + , 3- and commucomplex 3-dimensional Lie algebra with basis tation relations
x3,
is isomorphic to Y3. I n addition the following isomorphisms Clearly, are easily established: for all positive integers Lemma 9.1 gP,*E 5??q,P; 9’np,np gP,*; P, 9, n. According to this lemma every Lie algebra 5??p,,,, is isomorphic to a Lie algebra gP,*such that (1) p and q are relatively prime positive integers; (2) p is odd; and (3) if q is odd then p > q. Consequently, from now on the pair ( p , q) will be assumed to satisfy properties (1)-(3). It is , 5??P‘,q’ with subscripts satisfying obvious that two Lie algebras gp,, these properties, are isomorphic if and only if p = p‘ and q = 4’. Applying the techniques developed in Chapter 8 we can determine all by gd’s in one or two of the transitive effective realizations of Yp,, complex variables. Thus, from (8.26) there follows the realization 71
:
a
-
aY ’
epy,
e-qy;
r
=
3, k
=
2, s
= 0.
(9.39)
Every transitive effective realization of gp,,by gd’s in one complex variable is an element of OZ(ql). Similarly, by making use of Lie’s tables (Lie [I], Vol. 111, pp. 71-73),
9-7.THE LIE GROUP G,,Q
31 3
it can be shown that every transitive effective realization of 9p,q by gd’s in two complex variables is an element of 0l(sj), j = 1, 2, 3, where r=3, 6, :
--PZ,
a a az, + 4% az, ’
a -
Y
az, ’
=
3. k
r=3,
9-7
k=l, s=O; =
1,
k = l ,
s = 0;
(9.40)
s=O.
The Lie Group Gp,q
Corresponding to a pair of relatively prime positive integers ( p , q) such that p is odd, denote by Gp,Qthe 3-dimensional complex Lie group with elements
g ( h c, 7)’
b, c,
E
I%,
and group multiplication
It is easy to check that the multiplication is associative. Furthermore, e = g(0, 0, 0) is the identity element and g(--e-PTb, -@c, -T) is the unique inverse of the group element g(b, c, T). GP,*has a 4 x 4 matrix realization
(9.42)
where matrix multiplication corresponds to the group operation. Comparing (9.42) with (1.35) we see that Gl,l is isomorphic to T , . T h e Lie algebra of G,,* can be computed from either of the expressions (9.41) or (9.42), and is easily recognized to be isomorphic to Y P , ,. Indeed we can make the identification
where f * ,
y 3generate 9fp,qand satisfy the commutation relations (9.38).
31 4
9-8
9. SOME GENERALIZATIONS
Representations of gp,q
Let p be a representation of g’p,q on an abstract vector space V and let = J*, p(Y3)= J3, be operators on V . Clearly,
p(f*)
Note that the operator C p * q = (J+)*(J-)” = (J-)”( J+)q commutes , on V . For p irreducible we would with all operators p(01), 01 E gP,, expect CP,* to be a multiple of I. I n strict analogy with our study of the representation theory of F3 we will classify all representations p satisfying the properties: (i) p is irreducible (ii) Each eigenvalue of J 3 has multiplicity equal to one. There is a countable basis for V consisting of eigenvectors of J3.
(9.45)
We give without proof the results of the straightforward classification. satisfying (9.45) and Theorem 9.2 Every representation p of gP,, 0, on V , is isomorphic to a representation of the form for which Cp’q QP,*(w,m,) defined for w , m, E @ such that w # 0 and 0 Rc m, < 1. T h e spectrum of J3 corresponding to QP,*(w,m,) is given by 5’ = (m, + n: n an integer) and there is a basis i f m } , m E S , f0r.V such that
+
<
J-fm
= wfm-,
J+fm
= mfm
J3fm
,
= wfm+v,
C”.ym= (I+)*( J - ) ” f i n
.
(9.46)
= wp+,fm
QpPQ(w‘, mi) if and only if There exist isomorphisms QP,,(w, m,) m, = mi , w = w‘d, where d is a ( p g)th root of unity. I n accordance with the usual procedure, we can use the operators to construct a realization of the representation QP,*(m,m,). This can be done by introducing the change of variable z = e” in (9.39) and considering the operators
+
J3
=z
d
+ c1 ,
J+
=C~ZP,
J-
= c~z-‘J
(9.47)
in Ol(fl). Thus, let ”y; be the complex vector space of all finite linear combinations of the functions h,(x) = zn, n = 0, * l, &2, ..., and define operators J*, J 3 on rY; by (9.47) where c1 = m, , c2 = w . Define
9-8. REPRESENTATIONS OF gP,@
the basis vectorsf, of Vl byf,(z) runs over the integers. Then
J+fm
= ( u z p ) ~ "= wfm+n
1
=
h,(z) where m
31 5 =
m,
+ n and n
= ( w ~ - Z" ~ )= wfm-,
J-fm
,
(9.48)
c.ym = w ~ + " f m . . These equations agree with (9.46) and yield a realization of Qp,p(w,m,). T h e differential operators (9.47) (cl = m, , c2 = w ) induce a multiplier representation A of G,,* on the space GZl consisting of those functions f ( z ) which are analytic and single-valued for all x f 0. This multiplier representation is defined by operators A(g),g = g(6, c, T ) E Gp,* : [A(g)f](z) = exp[w(bzP
+ cz-@)+ r n o ~ ] f ( e T z ) ,
f~ & .
(9.49)
Clearly, GZl is invariant under the operators A(g) and the group property A(g1g2) = A(g,) A(g2) is valid for all g, g, E G,,* . T h e matrix elements A,,(g) of A(g) with respect to the analytic basis {f, = h,) of GZl are defined by 9
c m
[A(g) h k l b )
=
M g ) hdz),
g E GD,*>
k
= 0 , il, *2,...,
(9.50)
I=-m
or exp[w(bzP
where g
= g(6, c,
T).
+ c x q ) + (m,+ k )
a,
T]
zk = I=--m
A,,(g) zz
(9.51)
Explicitly, (9.52) (9.53)
T h e nonnegative integers sl ,n l are uniquely determined by the properties:
(1) 1 = n,p - siq, (2) If I = zip - sip where n ; , s; are nonnegative integers, then n;
+ si 3 n, +
s,.
Since p and q are relatively prime, the integers n, , s l can easily be shown to exist for all 1. For example, if p = q = 1 then s, = 0, n, = I for I 3 0 and s, = -1, n, = 0 for 1 < 0.
31 6
9. SOME GENERALIZATIONS
If bc # 0 we can introduce new group parameters r, v defined by b = r v p / ( p + p), c = r v - Q / ( p 9). I n terms of these new parameters the matrix elements are
+
(9.54) (9.55)
It follows from the ratio test that 17,q(r)is an entire function of r for all integers 1. Here I;,’(r) = (-i)zJz(ir), 13 0, is the ordinary “modified Bessel function” (see ErdClyi et al. [l], Vol. 11). T h u s we can consider the functions If’*(r) to be a group-theoretic generalization of Bessel functions. Substitute (9.54) into (9.51) to obtain the simple generating function Y
(z”
+ z-.)I
f
=
IpqY)
2.
(9.56)
Z=-m
T h e group property of the operators A(g) implies the addition theorem
c Azj(g1) A&!Z), m
Azk(glg2) =
(9.57)
j=-W
valid for all g, ,g,
E
Gp,q. This leads immediately to the identity
+ b, , + cZ) = 2 F;:q(b, m
FP.*(bl
~1
j=-m
, c ~ ) F ; ” (, c2). ~~
(9.58)
We could also use the addition theorem to derive identities involving the functions Iy3q(r).However, this will not be done here as most of the results so obtained are special cases of identities which will be derived in Section 9-10.
9-9
Recursion Relations for the Matrix Elements
Denote by GZ(Gp,J the space consisting of all entire functions on the group Gp,*, i.e., of all entire functions of the parameters b, c, T. Clearly, the matrix elements A,,(g) are elements of 6Y(Gp,J. T h e representation P of Gp,qon LT(Gp,J defined by [P(g’)fl(g) = f(gg’),
f
.
U(GP,*)Y
(9.59)
317
9-9. RECURSION RELATIONS FOR MATRIX ELEMENTS for all g, g‘ E Gp,qsatisfies the relation
and determines Gp,qas a transformation group. According to (9.57) the action of P on a matrix element Ajk is given by
c m
[P(g‘)Ajkl(g) = Ajk(gg’) =
Adg’) M g ) .
(9.60)
l=-m
Comparing this expression with (9.50) we find that for fixed j the functions (Aik},K = 0, f l , f 2 , ..., form a basis for a realization of the representation Qp,,(w, m,) of Gp,qunder the action of the operators P(g).Therefore, the Lie derivatives J*, J 3 defined on a ( G p , Jby
(9.61)
JYk) =
d
I
[ P ( ~ x~$~)f’l(g) P
.r=O
f~ @(Gp,Q),
satisfy the relations J3A9&)
=
(%
4- k) A,&), J-Ajdg)
(J’)“ (J-)” A,,(g)
CP*QAJk(g)=
=
=
J+Aidg)
WAili+pk),
“A,,-&),
wYfyAjk(g)>
(9.62)
i,k
=
0, & I >
*’>***
*
Equations (9.62) yield recursion relations and differential equations for the matrix elements A j , . We will use these equations to derive recursion relations for the generalized Bessel functions I ~ , * ( I Thus, ). we assign to a group element g = g(b, c, T ) , bc # 0, the local coordinates [ I , u, T], where b = rvp/(p q), c = ~v-q/i(p 9). Then if g‘ = g(b‘, c’, T‘), a straightforward computation shows the local coordinates of gg‘ are
+
[Y(
1
v(1
+ ( p + q)
r-lv-PeW)q/(P+y)
+
(1 + ( p f q ) y-laG‘e-YTC’
+ ( p + q ) r-lv-PePTb’)ll(P+y)(1
i
1
Y/(P+Y)
7
( p + q ) Y-~vQ~-RTc’)~/(P+*), T + T’]
for 1 b’ 1, 1 c‘ I sufficiently small. It follows from the definition (9.61) that the Lie derivatives J*, J 3 are
(9.63)
31 8
9. SOME GENERALIZATIONS
Substituting (9.54) and (9.63) into (9.62) and factoring out the dependence on v and T, we obtain the equations
= I$*(r),
m
=
0, i l , *2
,... .
(9.64)
I n particular, the functions Ig*((r)are solutions of an ordinary differential q. equation of order p
+
9-10
Realizations in Two Variables
T h e operators 6, and 6, , (9.40) can be used to construct realizations of
Qp,,(w, m,) on spaces of functions of two complex variables. However, the basis functions so obtained are elementary and of little interest. On the other hand a realization of Qp,q(w,m,) by 6, does lead to new results. T o see this we introduce new variables x , y , where z1= xePY/(p q), in the expression for 6, :
z2 = xe-""/(p
+
+ q),
T o construct a realization of Qp,q(w,m,) using the operators (9.65) we must find nonzero functions &(x, y ) = Z,(x) e c r n y ; m = m, n, n = 0, f l , 5 2 ,..., such that expressions (9.46) are valid. These expressions are equivalent to the equations
+
d
) ... (q dx X
dx
m-qp X
)(p
d
z
m-(p-l)q +
X
1
(9.67)
where without loss of generality for special function theory, we have set w = 1. Thus, the functions Z,,L(x)are solutions of differential equations of order p + q.
9-1 0. REALIZATIONS IN TWO VARIABLES
319
I n general it is possible to findp + q linearly independent solutions of Eqs. (9.66) and (9.67). However, we will be content with a single solution. It is a consequence of (9.64) that if m, = 0, the functions Z,(x) = I ? ) ~ ( Qare ( x )solutions of these equations. Following a standard procedure in the theory of special functions we shall use analytic ( Xm) an arbitrary complex number and continuation to define I ; L ~for show that the functions Z,(x) = Z?)$(x) satisfy the required relations (9.66) and (9.67) even if m, f 0. An application of the Cauchy residue theorem to (9.56) yields 27riI:q(r)
=
s,
z-”-l exp
[ z (z” + 2-91 dz, P+4 m
= 0,f l ,
*2 ,...,
where C is a simple closed contour surrounding the origin in the z-plane. Without changing the value of the integral, the contour C can be deformed into a loop D which starts at infinity on the negative real z-axis, encircles the origin counterclockwise and returns to its starting point. ( T he proof of this statement is almost word for word the same as the well-known proof of the analogous statement in the special case p = q = 1, see Whittaker and Watson [I], Chapter 17). Since p is odd, the integrand is bounded on D for Re Y > 0. Thus, we have
s‘”’
where --m denotes integration along D. Since the integrand is now single valued on D even for m complex, we can use (9.68) to define the function IPg(r) for arbitrary values of m and Re Y > 0. Moreover, by differentiating under the integral sign in (9.68) we can verify that Eqs. (9.64) remain valid for arbitrary m. T h u s the functions Z,,(x) = IP;;(x), m = m, n, define a realization of the representation Q p , q ( m o , 1) of %I, ., We can obtain more information about the generalized Bessel functions *;:Z by mimicking the standard treatment of ordinary Bessel functions in terms of contour integrals as given, for example, by Whittaker and Watson [I], Chapter 17. If r > 0, introduction of the new variable w = ( ~ / ( p+ q ) ) l / p z in (9.68) leads to the expression
+
T h e right-hand side of this equation can be analytically continued to define I:iq(y) as an analytic, but not single valued, function of r for all
320
9. SOME GENERALIZATIONS
r # 0. I n particular r-m/pI$g(r) is an entire function of r ( P + q ) / P . T o obtain a series expansion for Zgp(r) we use
in (9.69) and integrate term by term. From the theory of the gamma function (Whittaker and Watson [l], Chapter 12),
Thus,
For p = 1 this expression simplifies to
We could construct additional solutions of equations (9.64), linearly independent of the functions IEQ(r),by choosing a different path of integration in (9.68) (see Khriptun [I]). However, for purposes of illustration we will be content with the solutions ZZ*(r).
9-1 I
Generating Functions for Generalized Bessel Functions
+
As shown above, the functions fm(x, y ) = I%(.) e c r n y , m = m, n, n an integer, form a basis for a realization of the representationQ,,q( 1, m,) of gP,,. Following our usual procedure we will extend this representation to a local multiplier representation of Gp,q in the space F of functions analytic i n ' a neighborhood of (xo,y o ) = (1, 0). T h e operators T(g),g = g(b, c , T ) E Gp,p defining this multiplier representation can easily be computed from (9.65) and (9.43). T h e result is
fE F ,
(9.72)
9-11. GENERATING FUNCTIONS
321
+
+
valid for j b(p q)/xtp I < 1, 1 c(p q) tq/x I < 1, where t = e”. The matrix elements of T(g) with respect to the analytic basis (fmfx, t ) = P Z ( x ) t-”), are the functions A,,(g) given by (9.52) and (9.54). Thus, m
[T(g)frn,,+kI(x,t )
C
=
= 0,
Azk(g)fm,+Z(X, t ) ,
i l , +2,-*.,
Z=-CC
or b(p
m
+ q))Q:(P+Q)
(1
+
c(p
+ q) tQ)n.(P+Q)] X
c F!;Q(b, m
=
c ) I2$x)
Z=--a,
I b(p for all m
E
$?.If c “ : 1
[X
tl,
+ q ) W I < 1,
=
0, t
(1
+ y-b(p )Qf + q )
=
I .(p
t q) tQ/X I
< 1,
(9.73)
1, this expression simplifies to / ( P + d ](1
+ b(p + q )
)rn/(P+Q)
X
while for b
=
0, t
I:.[+
=
1 it becomes
+
c(p
+ ,I))”’(”+Q)] (
+
c(p
+ q))-”L;(”+Q) X
+
If b = c = ri(p Q ) # 0 the matrix elements can be expressed in terms of generalized Bessel functions and (9.73) becomes y
e Q I X ( l
Q/(P+Q)
+,t”)
) (1, +
rtg
pl(P+g)
I(
x
+ rt-”
x f rtQ
1
~/(P+Q!
For p = Q = 1, Eq. (9.73) is equivalent to the Bessel function identity (3.29).
322
9. SOME GENERALIZATIONS
Similarly, in analogy with Weisner’s treatment of Bessel functions as given in Section 3-4, we can use the operators
expressions (9.65), to derive generating functions which are not related to the matrix elements A,,(g). T h e basic observation to be made is: If f ( x , t ) is a solution of the equation C p , v = ( J + ) q ( J - ) p f = wf then the function T(g)f given by the formal expression (9.72) is also a solution of this equation. Moreover, iff is a solution of the equation (xIJ+
+ xzJ- + x,J’)lf(x,
t ) = xf(x, t )
for constants x1 , x2 , x3 , A, then T(g)f is a solution of
As an illustration of these remarks, consider the function f ( x , t ) = I z * ( x )tm, m E @. In this case C p , q f = f, JY = -mf, so the function T(g)f satisfies
where g = g(b, c, T) E G p , q .If c in the form
=T =
0, b
Since x - ” / p I E , q ( x ) is an entire function of expansion in t P about t = 0:
cm
h(x, t )
=
n=-m
h,(x) tn”,
=
1, T(g)f can be written
x(p+q)lp,
I X t Y I
h has a Laurent
+ q.
323
9-11. GENERATING FUNCTIONS
Substituting this expansion into the first equation (9.74), we find Z,,(x) = h-,(x) is a solution of the generalized Bessel equation (9.67) for n = 0, 4 1 , 4 2 , ... . At this point we use the fact: For m an integer the only solutions of (9.67) which are regular at x = 0 are cI?:(x), c a constant. This statement can be verified by computing the indicia1 equation of (9.67) (see Ince [I], Chapter 7). Since h(x, t ) is bounded for x = 0, we have h,(x) = c,I&'(x), c, E @. Hence,
c m
h(x, t )
=
n=-m
C " l g ( x ) t".
+
T h e equation (-p J+ J3) h(x, t ) = -mh(x, t ) implies = ( m - np) c, . T h e constant co can be determined by setting x = 0 in h(x, t ) and using (9.70). Consequently, co =
T ( - m / p ) sin[r(m Pr
+ 1)1 ,
cn =
(-1)" T[(.P
-
mYP1 sin[r(m
+ 111
Pn
and we obtain the identity
139"
When
p
=
Ic p
+ q.
q = 1 this expression is equivalent to (3.38).
(9.76)
Appendix
For reference we list here some fundamental properties of special functions which occur frequently throughout the book. All of these properties can be found in Magnus et al. [I] and, of course, in ErdClyi et al. [l]. I n the following expressions p, v and 5 are arbitrary complex numbers and n is an integer.
1 The Gamma Function For Re z
> 0 the function T ( z )is defined in terms of the integral
However, by analytic continuation r ( z ) can be extended to a function analytic in the whole complex plane, with the exception of simple poles at z = --n, n = 0 , 1 , 2 ,.... Functional equations: T(z
+ 1) = z r ( Z ) ,
Special values:
Binomial coefficients:
T ( z )T ( l - z )
=
7r
.
sin 7rz
(-4.1)
2. THE HYPERGEOMETRIC FUNCTION
325
2 The Hypergeometric Function T h e hypergeometric series F(u, b ; C;
Z) =
=
,J1(a, b; C;
Z)
ab z +--+ c I! + + 1)
1
+ 1)
-
2! ( U + l~ - 1) b(b 1) (b c(c 1) - * * (c n - 1) c(c
U(U
+
+ 1)b(b + 1) z2 + ...
a(a
+
...
+ +
'1.
+ n - 1) zn n!
(A-4) defines a function which is analytic when 1 z 1 < 1. If a or b equals -n, n = 0, 1 , 2, ..., the series is finite. F(a, b ; c; x) can be analytically continued to define a function analytic and single-valued throughout the cut x-plane, where the cut runs along the positive real axis from the branch point +1 to +a.For Re c > Re b > 0 this function has the integral representation
('1
F ( a , b; c; 2) =
v ) r ( c
Limit relation: lim Ffa, b; C; z )
r(c)
~
c+-n
tb-l(l
b)
~
a(a
- t)c-b-l
+ 1) -.. + n) b(b + 1) (U
~
(n
F(a, b ; c ; x) is a solution, regular at z equation d2u
For
c
xl-CF(a
# 0, -1, -
c
+ l)!
. zn+lF(a + n + 1, b n = 0 , 1 , 2 ,....
~ (1 Z) p
+
[C
~
(1
(U
-
tz)-" dt.
0
(b
+ n)
+ n + 1; + 2; z), ('4.5)
0, of the hypergeometric
=
+ b + 1)
-.*
Z]
du --~ b u = 0. dz
-2,...,this equation has
+ 1, b + 1 ; c + 1 ; x).
a
(A4
second
solution
Differential recursion formulas: d
+ 1, b + 1; c + 1; z ) F ( a , b; c; z ) = aF(a + 1, b; c; Z)
--(a,
dz
+ z dz -1d
[a L
[(a - c)
[(a
+ bz
+b
-
z(1
- c) -
(1
~
ab b; c; 2) = -F("
"1 F(u, b ;
Z) -
dz
C
C;
X) = ( a
(b
-
~
c ) F ( u - 1, b ; C; C)(C C
(C -
-U)
Z)
F ( a , b; c
1) F(u, b ; c
-
1; Z)
(A.7)
+ I ; Z)
APPENDIX
326 Transformation formulas:
=
(1 - z)C-"-bF (c - U , c
(A.8)
b ; C; z).
-
Some functions represented by hypergeometric functions: (i)
Legendre polynomials P,(cos 0)
(ii)
=F
(n + 1, - n ;
,
1;
n
0, 1, 2 ,... .
(A.9)
General spherical harmonics %;(z)=
r(l - p )
(ZLL)P'2F (-", + 1; 1 z 1 -
z+l
'
2
-
-9 v
(arg(z2 - 1) arg z
(iv)
=
- p;
3 1 + 3; 7)
for z real and > 1,
0, for z real and
Gegenbauer polynomials q t )=
+
r ( n 2v) F (n r ( n I) W v )
+
1.
0 for z real and > 1
= 0,
=
2
+ 2v, -n; v + t;
2
> 0).
3. THE CONFLUENT HYPERGEOMETRIC FUNCTION
(v)
Jacobi polynomials 2) = F(--n,
%(a, y ,
01
+ n ; y ; x),
327
n = 0, 1 , 2,...
3 The Confluent Hypergeometric Function T h e confluent hypergeometric series $-,(a; c ;
z)
=
+ 1) + ... + a-c Iz! + a(a c(c + 1) 2! a ( a + 1) .*.( a + n 1) zn + ...
1
z2 -
-
-
+
c(C+l)~..(C$n-l)
'
n!
(A.lO)
defines an entire function of z. This function is a solution of the differential equation d'v dz2
z -+ (c
-
dV
z ) - - av
= 0.
dz
(A.ll)
For c not an integer, a second solution of (A. 11) is z1-c
,F,(a
-c
+ 1; 2
-
c;
z).
Limit relation:
n
= 0,
1 , 2,... .
(A.12)
Recursion formulas: d
& ,F,(a;
[-1 [(c
-
-1 -1
+ dzd
,F,(a;
c;
z)
=
c;
z)
=
d 1) 1z ,F,(a; c ; z ) dz
a
;,F,(a a-c ~
C
= (C
-
1) ,F,(a;
z)
=
a ,F,(a
[-z-t(c--)+a-] d ,F,(a; c ; z ) dz
=
(c
,F,(a;
c;
-
+ 1; c + 1; z ) ,F,(a; c + 1; z ) , c
-
1; z),
(A.13)
+ 1; c ; z ) ,
a ) ,F1(a - 1; c ; z).
Transformation formula: ,F1(a; c ; z )
= ez
,F,(c
-
a ; c ; -z).
(A.14)
APPENDIX
328
Some functions represented by confluent hypergeometric functions: (i)
Generalized Laguerre functions
(ii)
Bessel functions (A. 15)
(iii) Parabolic cylinder functions
4 Parabolic Cylinder Functions T h e parabolic cylinder function Dy(z),(A.15),(iii), is a solution of the differential equation d2u p + (u
+ 51 - -$)
u =0
(A.16)
as are the functions D y ( - z ) , D--y--l(iz),D-.-l(-iz). Here Dy(z) and D-u-l(iz) are linearly independent for all Y, and if u is not an integer D y ( z )and D y ( - z ) are linearly independent. If v = n is a nonnegative integer then &(z) = 2-n/2exp(-z2/4) Hn(2-'l2z), (A.17)
where H,(z)
=
dn (- 1)" exp(z2)-exp( -2)
is the Hermite polynomial of order n. Recursion formulas:
dzn
(A. 18)
5. BESSEL FUNCTIONS
5
329
Bessel Functions
The Bessel function Ju(z)can be defined by the series expansion
+ 1; -z2/4),
$,(v Jdz) = r ( (z’2)y V + 1)
I arg z 1 < T ,
(A.20)
where $,(c;
z) = 1
1 + cl l!z + c(c _ _ _ - + ..* + 1)2! --
1
22
+ c(c
Zn
+ 1) ... (c + n - 1) n!
+
. . I .
(A.21)
Clearly, z-”JV(z)is an entire function of z. T h e Ju(z)is a solution of Bessel’s equation (A.22) as is JPu(z). For v not an integer, Jv(z)and J J z ) are linearly independent. However, if v = n is an integer then J-n(z) = (-l)n Jn(z)and Jn(z) is the only solution of (A.22) which is regular at z = 0. Recursion formulas:
Spherical Bessel functions: (A.24)
Bibliogruphy
AKIM,E., and LEVIN,A. [l] A Generating Function for the Clebsch-Gordan 629-631 (1961) (English Transl.).
Coefficients, Soviet Math. 2,
M., and GUNSON, J. ANDREWS, El] Complex Angular Momentum and Many-Particle States. I. Properties of Local Representations of the Rotation Group, /. Math. Phys. 5, 1391-1400 (1964). BARGMANN, V. [l] Irreducible Unitary Representations of the Lorentz Group, Ann. of Math. 48, 568-640 (1 947). [2] On Unitary Ray Representations of Continuous Groups, Ann. of Math. 59, 1-46 (1954). [3] On a Hilbert Space of Analytic Functions and an Associated Integral Transform, I. Comm. Pure Appl. Math. 14, 187-214 (1961). [4] On the Representations of the Rotation Group, Rev. Mod. Phys. 34,829-845 (1962); also in “Quantum Theory of Angular Momentum” (L. C. Biedenharn and H. Van Dam, eds.), pp. 30&316. Academic Press, New York, 1965. C. BARUT,A. O., and FRONSDAL, [I] On Non-Compact Groups. 11. Representations of the 2 ROY.SOC. A287, 519-531 (1965).
+ 1 Lorentz Group, PYOC.
G. BELTRAMETTI, E. G., and LUZATTO, [l] Rotation Matrices Corresponding to Complex Angular Momenta, Nuovo Cimento 29, 1003-1018 (1963).
L. C., and VANDAM,H. (eds.) BIEDENHARN, [l] “Quantum Theory of Angular Momentum.” Academic Press, New York, 1965. BINGEN,F. [I] Les Fonctions de Bessel d’ordre Entier et Les ReprCsentations du Groupe des DCplacements du Plan, Bull. SOC.Math. Belg. 17, 115-152 (1965).
H. BOERNER, [ 11 “Representations of Groups,” North-Holland, Amsterdam, 1963. CARTAN, H., and EILENBERG, S. [l] “Homological Algebra.” Princeton Univ. Press, Princeton, New Jersey, 1956.
BIBLIOGRAPHY
331
CHEVALLEY, C. [1] “Theory of Lie Groups.” Princeton Univ. Press, Princeton, New Jersey, 1946. COHN,P. M. [l] “Lie Groups.” Cambridge Univ. Press, London and New York, 1957. COURANT, R., and HILBERT,D. [l] “Methods of Mathematical Physics,” Vol. 1 . Wiley (Interscience), New York, 1953.
A. 2. DOLGINOV, [l] Relativistic Spherical Functions, Soviet Phys. JETP 3, 589-596 (1956) (English Transl.). DUNFORD, N., and SCHWARTZ, J. T. [l] “Linear Operators,” Part I. Wiley (Interscience), New York, 1958. EISENHART, L. P. [l] “Continuous Groups of Transformations.” Dover, New York, 1961. A,, MAGNUS, W., OBERHETTINGER, F., and TRICOMI, F. ERDELYI, [l] “Higher Transcendental Functions” (Bateman Manuscript Project), Vols. I, 11. McGraw-Hill, New York, 1953.
J. FRIEDMAN, B., and RUSSEK, [l] Addition Theorems for Spherical Waves, Quart. A p p l . Math. 12, 13-23 (1954). FURSTENBERG, H. [l] Translation-Invariant Cones of Functions on Semi-Simple Lie Groups, Bull. Amer. Math. SOC.71, 271-326 (1965). GARDING,L. [l] Vecteurs Analytiques dans les Representations des Groupes de Lie, Bull. SOC. Math. France 88, 73-93 (1960). I. M., and GRAEV, M. GEL’FAND, [I] Representations of a Group of Matrices of the Second Order with Elements from a Locally Compact Field and Special Functions on Locally Compact Fields, Russian Math. Surveys 18, No. 4, 29-100 (1963) (English Transl.).
I. M., and NAIMARK, M. A. GEL’FAND, El] “Unitary Representations of the Classical Groups” (Trudy Mat. Inst. Steklov. 39), Moskov-Leningrad, 1950 (German Transl., Akademie-Verlag, Berlin, 1957). GEL’FAND, I. M., MINLOS,R. A., and SHAPIRO, 2. Y. [I] “Representations of the Rotation and Lorentz Groups and Their Applications.” Macmillan, New York, 1963 (English Transl.).
I. M., GRAEV, M. I., and VILENKIN, N. Y. GEL’FAND, [2] “Generalized Functions,” Vol. 5 , Academic Press, New York, 1966 (English Transl.). GODEMENT, R. [I] A Theory of Spherical Functions, I, Trans. Amer. Math. SOC.73, 496-556 (1952). GUGGENHEIMER, H. W. [l] “Differential Geometry.” McGraw-Hill, New York, 1963.
3 32
BIBLIOGRAPHY
GUNNING, R. C., and ROSSI,H. [l] “Analytic Functions of Several Complex Variables.” Prentice-Hall, Englewood Cliffs, New Jersey, 1964. HALMOS, P. [l] “Finite Dimensional Vector Spaces,” 2nd ed. Van Nostrand, Princeton, New Jersey, 1958. HAMERMFSH, M. [l] “Group Theory and Its Applications to Physical Problems.” Addison-Wesley, Reading, Massachusetts, 1962. HELGASON, S. [l] “Differential Geometry and Symmetric Spaces.” Academic Press, New York, 1962. HERMANN, R. [l] Compactifications of Homogeneous Spaces and Contractions of Lie Groups, Proc. Natl. Acad. Sci. U.S. 51, 456461 (1964). [2] “Lie Groups for Physicists.” Benjamin, New York, 1966. INCE,E. L. [l] “Ordinary Differential Equations.” Dover, New York, 1956. INFELD,L., and HULL,T. [l] The Factorization Method, Reos. Mod. Phys. 23, 21-68 (1951). JACOBSON, N. [l] “Lie Algebras.” Wiley (Interscience), New York, 1962. KAUFMAN, B. [l] Special Functions of Mathematical Physics from the Viewpoint of Lie Algebra, J. Math. Phys. 7 , 447457 (1966). KHRIPTUN, M. [I] On an Ordinary Differential Equation of Higher Order, Ukrain. Mat. Z . 15, 277-289 (1963) (in Russian). KUNZE,R. A., STEIN, E. M. [l] Uniformly Bounded Representations and Harmonic Analysis of the 2 x 2 Real Unimodular Group, Amer. J. Math. 82, 1-62 (1960). KUR$UNO~LU, B. [I] “Modern Quantum Theory.” Freeman, San Francisco, 1962. LANDAU, L., and LIFSHITZ, E. [l] “Quantum Mechanics.” Addison-Wesley, Reading, Massachusetts, 1958. LIE, S. [l] “Theorie der Transformationsgruppen,” Vols. I, 11,111. Unter Mitwirkung von. F. Engel., Leipzig, 1888, 1890, 1893.
G. Y. LYUBARSKII, [l] “The Application of Group Theory to Physics.” Pergamon Press, Oxford, 1960 (English Transl.). MACKEY, G. W. [l] The Theory of Group Representations, Univ. of Chicago lecture notes (1955).
BI BLI OG RAPHY
333
MAGNUS, W., OBERHETTINGER, F., and SONI,R. [I] “Formulas and Theorems for the Special Functions of Mathematical Physics,” 3rd ed. Springer-Verlag, New York, 1966. MILLER,W. [l] On Lie Algebras and Some Special Functions of Mathematical Physics, A M S Mern., No. 50, Providence (1964). [2] Some Applications of the Representation Theory of the Euclidean Group in ThreeSpace, Comm. Pure Appl. Math. 17, 527-540 (1964). [3] On a Generalization of Bessel Functions, Comm. Pure Appl. Math. 18, 493-499 (1 964). [4] On the Special Function Theory of Occupation Number Space, Comm. Pure Appl. Math. 18, 679-696 (1965). [5] On the Special Function Theory of Occupation Number Space, 11, Comm. Pure APPZ. Math. 19, 125-138 (1966). MURRAY, F. J., and VONNEUMANN, J. [I] On Rings of Operators, Ann. of Math. 37, 116-229 (1936). NAIMARK, M. A. [I J “Normed Rings.” Noordhoff, Groningen, The Netherlands, 1959 (English Transl.). [2] “Linear Representations of the Lorentz Group.” Macmillan, New York, 1964 (English Transl.). NELSON, E. [l] Analytic Vectors, Ann. of Math. 70, 572-615 (1959). PAULI,W. [l] Continuous Groups in Quantum Mechanics, Ergeb. Exakt. Naturw. 37, 85-104 (1965). PETROVSKY, I. G. [I] “Lectures on Partial Differential Equations.” Wiley (Interscience), New York, 1954 (English Transl.). PONTRJAGIN, L. S. [I] “Topological Groups.” Princeton Univ. Press, Princeton, New Jersey, 1958. PUKANSZKY, L. [I] On Kronecker Products of Irreducible Representations of the 2 X 2 Real Unimodular Group, Part I, Trans. Amer. Math. SOC.100, 1, 116-152 (1961). REGGE,T. [l] Symmetry Properties of Clebsch-Gordan’s Coefficients, Nuowo Cimento [lo] 10, 544-545 (1958). ROMM,B. D. [l] Decomposition into Irreducible Representations of a Tensor Product of Two Irreducible Representations of the Real Lorentz Group (The Case of Two Discrete Series). Zzu. AKad Nauk, SSSR, Ser. Math. 28, 855-866 (1964) (in Russian). ROSE,M. E. [I] “Elementary Theory of Angular Momentum.’’ Wiley, New York, 1957.
334
BlBLlOG RAPHY
SALETAN, E. J. [I] Contraction of Lie Groups, J. Math. Phys. 2, 1-21 (1961). SCHWINGER, J. [I] On Angular Momentum, in “Quantum Theory of Angular Momentum” (L. C. Biedenharn and H. Van Dam, ed.), pp. 229-279. Academic Press, New York, 1965. SEGAL, I. E. [I] Lectures at the Summer Seminar on Applied Mathematics (unpublished), Boulder, Colorado, 1960. SEMINAIRE “SOPHUS LIE” [l] “Thborie des Algkbres de Lie, Topologie des Groupes de Lie.” Paris, 1955. SHARP,W. T. [l] Racah Algebra and the Contraction of Groups, Research Rept., CRT-935, AECL1098, Atomic Energy of Canada Limited, Chalk River, Ontario, 1960.
TRUESDELL, C. [l] “A Unified Theory of Special Functions.” Princeton Univ. Press, Princeton, New Jersey, 1948. VANDER WAERDEN, L. [l] “Gruppentheoretische Berlin, 1932.
Methode
in der Quantenmechanik.” Springer-Verlag,
VILENKIN, N. Y. [l] Bessel Functions and Representations of the Group of Euclidean Motion, Uspehi. Mat. Nauk [N.S.], 11, No. 3 (69), 69-112 (1956) (in Russian). [2] Special Functions Connected with Class I Representations of Groups of Motions in Spaces of Constant Curvature, Trans. Moscow Math. SOC.1963, 209-290, AMS, Providence (1965) (English Transl.). [3] “Special Functions and Theory of Group Representations.” Izd. Nauka., Moscow, 1965 (in Russian); “Special Functions and the Theory of Group Representations.” AMS Transl., Providence, Rhode Island, 1968 (English Transl.). VILENKIN, N. Y., AKIM,E. L., and LEVIN,A. A. [l] The Matrix Elements of Irreducible Unitary Representations of the Group of Euclidean Three-Dimensional Space Motions and Their Properties, Dokl. Akad. Nauk. S S S R 112, 987-989 (1957) (in Russian). WEISNER, L. [l] Group-Theoretic Origin of Certain Generating Functions, Pacific J. Math. 5, 1033-1039 (1955). [2] Generating Functions for Hermite Functions, Can. /. Math. 11, 141-147 (1959). [3] Generating Functions for Bessel Functions, Can. /. Math. 11, 148-155 (1959). WEYL,H. [I] “Theory of Groups and Quantum Mechanics.” Dover, New York, 1955. [2] “The Classical Groups.” Princeton Univ. Press, Princeton, New Jersey, 1946. WHITTAKER, E. T., and WATSON, G. N. [ l ] “Modern Analysis,” 4th ed. Cambridge Univ. Press, London and New York, 1946.
BI BLI OG RAPHY
335
WIGHTMAN, A. S. [l] On the Localizability of Quantum Mechanical Systems, Rev. Mod. Phys. 34, 845-872 (1962). WIGNER,E. P. [l] On Unitary Representations of the Inhomogeneous Lorentz Group, Ann. Math. 40, 149-204 (1939). [2] The Application of Group Theory to the Special Functions of Mathematical Physics, Princeton lecture notes (1955). [3] “Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra.” Academic Press, New York, 1959. WIGNER,E. P., and INONU,E. [l] On the Contractions of Groups and Their Representations, Proc. Nut. Acad. Sci. U S . 39, 510-524 (1953). ~ E L O B E N K O ,D.
[l] The Theory of Linear Representations of Complex and Real Lie Groups, Trans. Moscow Math. SOC.1963, 57-110; AMS, Providence, Rhode Island, 1965 (English Transl.).
Numbers in italics refer to pages on which the references are listed.
Addition theorem(s), 22, 29 for spherical waves, 264 Akim, E. L., 242, 330, 334 Analytic basis, 28 Andrews, M., 330 Angular morentum operators, 33 Automorphism, see Homomorphism Bargmann, V., 118, 137-139,141, 142, 1 75, 222,230,232,234,236,238,330 Barut, A. O., 330 Bessel functions, 48, 51-77, 145, 150, 153. ' 192, 241, 328, 329 generalized, 316, 319 Beltrametti, E. G., 330 Bingen, F., 77, 330 Boerner, H., 6, 330 Cartan, H., 288, 330 Casimir operator, 32 Chevalley, C., 331 Clebsch-Gordan coefficients, 98, 129, 134, 149, 174, 178, 221, 263 Cohn, P. M., 1, 279, 331 Cohomology classes, 281-297 Commutator, 3, 13 subgroup, 286 Complexification of Lie algebra, 66 Confluent hypergeometric functions, 47, 78-153,183-194,212,254,304,327 Contraction of Lie algebras, 37, 150, 238 Courant, R., 138, 331
Derivation, see Generalized derivation Dolginov, A. Z., 331 Dunford, N., 26, 331 Eilenberg, S., 288, 330 Eisenhart, L. P., 2, 331 ErdClyi, A., 36, 63, 107, 198 270, 316, 324, 33I Euclidean group in plane, 65-77 in 3-space, 255-266 Euler coordinates, 256 Exponential map, 5 , 10, 19 Factorization method, 267-276 Factorization types, 46-48, 272-276 A, 194 B, 183 C', 106 C ' , 59 D',101 D , 65 E, 249 F, 253 Friedman, B., 264, 331 Fronsdal, C., 330 Furstenberg, H., 331 Gamma function, 324 Garding, L., 331 Gegenbauer polynomials, 202, 252, 326 Gel'fand, I. M., 36, 172,215, 220-222,331
INDEX General linear group, 5 Generalized derivation, 278 Generating function, 29 Godement, R., 331 Graev, M., 222, 331 Graf’s addition theorem, 63, 75 Guggenheimer, H. W., 1, 331 Gunning, R. C., 28, 278, 332 Gunson, J., 330 Haar measure, 123, 220 Halmos, P., 26, 332 Hamermesh, M., 1, 31, 175, 214, 221, 332 Helgason, S., 2, 66-69, 332 Hermann, R., 332 Hermite functions, see Parabolic cylinder functions Hilbert, D., 138, 331 Hilbert space, 67, 117 Hille-Hardy formula, 193 Homomorphism, 3 , 4 Hull, T., 46, 268, 274, 332 Hypergeometric functions, 47, 154-241, 250, 325 Ince, E. L., 323, 332 Infeld, L., 46, 268, 274, 332 Inonu, E., 38, 238, 335 Isomorphism, see homomorphism Isotropy subgroup, 285 subalgebra, 287 Jacobi polynomials, 100, 131, 164, 205, 211, 241, 327 Jacobson, N., 39, 252, 277, 282, 283, 288, 289. 332 Kaufman, B., 332 Khriptun, M., 320, 332 Kunze, R. A., 332 Kurgunoglu, B., 263, 332 Laguerre functions, see Confluent hypergeometric functions Landau, L., 34, 332 Legendre functions, 198 Levin, A. A,, 242, 330, 334 Lie algebra, 3, 25 finite-dimensional, 3
337
Lie derivatives, 13 generalized, 17, 30 linearly independent, 15 Lie group, 1 complex, 5 real, 5 Lie, S., 1, 278, 281, 290, 293, 295, 299, 312, 332 Lie transformation group, 12-24 effective, 14, 18 Lifshitz, E., 34, 332 Linear operator, 26 Lommel’s formulas, 62 Luzatto, G., 330 Lyubarskii, G. Y., 175, 332 Mackey, G. W., 266, 332 Magnus, W., 36,60,63, 107, 198,270,316, 324, 331, 333 Matrix elements, 22, 29, 73 Mehler’s formula, 140, 311 Miller, W., 242, 266, 333 Minlos, R. A., 36, 172, 215, 220, 221, 331 Multiplier representation, 17 isomorphic representations, 19 Murray, F. J., 146, 333 Naimark, M. A., 67, 69, 123, 137, 221, 246, 262, 331, 333 Nelson, E., 333 Oberhettinger, F., 36, 60, 63, 107, 198, 270, 316, 324, 331, 333 Occupation number space, 117 Parabolic cylinder functions, 48, 102-106, 138, 304, 328 Pauli, W., 333 Peter-Weyl theorem, 77, 221 Petrovsky, I. G., 271, 333 Pontrjagin, L. S . , 1, 2, 6, 9, 11, 220, 280, 285, 333 Projection operator, 123 Pukinszky, L., 222, 238, 333 Real form of Lie algebra, 66 Realizations of groups and algebras, 27, 45, 279, 293 Recursion relations, 29, 244, 268 Regge, T., 333
338
INDEX
Regular realization, 290 Representation irreducible, 27, 39, 68 of Lie algebra, 25, 38 of Lie group, 27, 61 unitary, 67 Romm, B. D., 238, 333 Rose, M. E., 333 Rossi, H., 28, 278, 332 Rotation group, 31, 215, 256 Russek, J., 264, 331 Saletan, E. J., 38, 334 Schwartz, J. T . , 26, 331 Schwinger, J., 175, 334 Segal, I. E., 118, 334 Shapiro, Z. Y., 36, 172, 215, 220, 221, 331 Sharp, W. T., 38, 334 Simply connected group, 9 Soni, R., 60, 324, 333 Special functions, 30 Special linear group, 7, 156 Special unitary group, 214, 256, 262 Spectrum, 38 Spherical Bessel functions, 252, 263, 329 Spherical harmonics, 31, 34, 160, 198, 263, 326
Stein, E. M., 332 Tangent vector, 2 Tensor products of representations, 94, 127, 170, 221, 238 Tricomi, F., 36, 63, 107, 198, 270, 316, 324, 331 Truesdell, C., 252, 334 Type A-F operators, see Factorization types Van der Waerden, L., 334 Vector space, 25 Vilenkin, N. Y . , 51, 17, 19, 222, 242, 299, 331, 334 von Neumann, J., 146, 333 Watson, G. N., 319, 320, 334 Weisner, L., 65, 210, 298, 334 Weisner’s method, 63, 190, 210, 322 Weyl, H., 334 Whittaker, E. T . , 319, 320, 334 Wightman, A. S., 259, 266, 335 Wigner, E. P., 38, 51, 175, 238, 335 Zelobenko, D., 335
