Series on Soviet and East European Mathematics Vol. 6
The Double Mellin-Barnes Type Integrals and their Applications to Convolution Theory
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Series on Soviet and East European Mathematics Vol. 6
The Double Mellin-Barnes Type Integrals and their Applications to Convolution Theory
Nguyen Thanh Hai & S B Yakubovich Byelorussian State University
World Scientific Singapore • New Jersey • London • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 73 Lynton Mead, Totteridge, London N20 8DH
Series on Soviet and East European Mathematics THE DOUBLE MELLION-BARNES TYPE INTEGRALS AND THEIR APPLICATIONS TO CONVOLUTION THEORY Copyright © 1992 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
ISBN 981-02-0690-9
Printed in Singapore by JBW Printers & Binders Pte. Ltd.
PREFACE
This book presents new results on the theory of double Mel1inBarnes type integrals and their applications to convolution theory. This class of integrals is known as the H-function of two variables and in the most general case it was first introduced by R.G.Buschman in 1978.
In an attempt
to make the book self-contained,
paragraph
§1 of
Chapter I provides the necessary brief historical background material in the theory of simple and double Mellin-Barnes integrals. In Chapter I we give the definition and the main properties of the H-function of two variables in the general case. In paragraphs §3 — §4 we first present the complete function of discuss
solution of
the convergence
two variables.
various
fundamental
problem
of
the general
In the following paragraphs properties
of
the
contiguous relations, the double Mellin transform,
general
(§5—§8)
Hwe
H-function:
series representa
tions. In the last paragraph §9 of Chapter I in order to classify the H-functions of two variables we introduce the notion of characteristic which will be used in the following Chapters.
In Chapter II we introduce and study the H-function of two varia bles with the third characteristic and its special case — the G-function of two variables. These functions are particular cases of the general H-function
and
they
have
immediate
applications
for
studying
the
convolution theory later on. Here, besides the convergence theorems we give various properties, which are habitual only for these functions. The list of special cases of the G-function of two variables is obtained in §13.
In Chapter III we present the modern method to study the H- and G-integral
transforms
together
with
their
generalizations.
consider these transforms in the special space 9K~
v
Here
we
(L) which is very
convenient to obtain the inversion theorems and it allows us to describe the composition structure of the mentioned transforms. Various particu lar cases of the G-transform are given.
In Chapter IV we construct and study the general integral convolu tions involving the classical Laplace convolution as special case. It gives rather a simple method to obtain the integral convolutions for Mellin type transforms. Many examples of convolutions for various known transforms are given. Here are considered new applications of known convolutions to evaluation of series and integrals.
For the sake of convenience, we give author, subject and notation indices in the end of the book.
This graduate
book
is
students
transforms.
written in
the
primarily areas
of
for
teachers,
special
researchers
functions
and
and
integral
In this book research workers and users in the field of
special functions of two variables will find new fundamental information and its application to the convolution theory.
Many persons have made a significant contribution to this book, both directly and indirectly. Contribution of subject matter is duly acknowledged
throughout
the
text
and
in
the
bibliography.
We
are
especially thankful to Professors Robert G. Buschman of the University of Wyoming, USA, Hari M.
Srivastava
of
the University
of Victoria,
Canada, and Megumi Saigo of the University of Fukuoka, Japan, for their keen support throughout the subject of this book and for sending us relevant reprints and preprints of their works.
This book is written during the academic year 1 9 9 0 — 1 9 9 1 "Research
Scientific
Laboratory
of
Applied
Methods
of
at the
Mathematical
Analysis" of the Byelorussian State University, where both authors work. We are immensely indebted to Professor Oleg
I.Marichev,
who was bur
scientific supervisor, for his constant encouragement during the last decade, when we studied at the Byelorussian State University.
vi
Finally, we are pleased to thank Mrs. Dr. Lyudmila K.Bizyuk for reading
the
manuscript
and
for
suggesting
a
number
of
invaluable
improvements.
June 1991
Dr. Nguyen Thanh Hai
Byelorussian State University
Dr. Semen B.Yakubovich
Minsk-80, USSR
vii
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CONTENTS Chapter I.
General H-function of Two Variables
§ 1.
Historical background
1
§ 2.
Definition and notations
9
§ 3.
The convergence region of the general H-function of two variables
12
§ 4.
The H-function of two real positive variables
23
§ 5.
Simple contiguous relations for the H-function of two variables
47
§ 6.
Main properties for the H-function
51
§ 7.
The double Mel 1 in transform
§ 8.
Series representations for the H-function of two variables
§ 9.
54
57
Characteristic of the general H-function of two variables
Chapter II.
69
The H-function of two variables with the third characteristic
§ 10. Definition and notations
72
§ 11. Convergence theorems
75
§ 12. Reduction formulas for the H-function with the third characteristic
80
§ 13. The G-function of two variables and its special cases...89 § 14. The double Kampe de Feriet hypergeometric series
ix
103
Chapter III. One-dimensional H-transform and its composition structure § 15. Spaces 3H"1 (L) and JJT1 (L)
119
§ 16. One-dimensional H-transform in the spaces 3JT1 (L) and UK"1 (L)
129
§ 17. The G-transform and its special cases
142
§ 18. Composition structure of the H- and G- transforms
153
Chapter IV.
General integral convolutions for the H-transform
§ 19. Classical Laplace convolution and its new properties...162 § 20. General integral convolution: definition, existence and factorization property
170
§ 21. Typical examples of the general convolutions
181
§ 22. Case of the same kernels: the general Laplace convolution
190
§ 23. G-convolution and its typical examples
198
§ 24. Convolutions for some classical integral transforms.... 209 § 25. Modified H-convolution
225
§ 26. General Leibniz rules and their integral analogs
233
Bibliography
261
Author Index
279
Subject Index
285
Notations
291
x
CHAPTER I. GENERAL H-FUNCTION OF TWO VARIABLES §1. Historical background
In 1812 C. F. Gauss systematically discussed the series (a) (b)
I
(1.1)
(c)
[where (a) = a(a+l)...(a+n-1); (a)
1], which is of fundamental im
portance in the theory of the special functions. This series is known as the Gauss series and it is represented by the symbol
The function
F
F (a,b;c;x).
and its various particular cases have already been
examined to considerable extent by a number of eminent scholars, notably C.F.Gauss, E.E.Kummer, S.Pincherle, H.J.Mellin,
E.W.Barnes, L.J.Slater,
Y.L.Luke and A.Erdelyi. A natural generalization of function, the so-called
F
is the generalized hypergeometric
F , which is defined in the following manner p q
(a)
F p q|
F
(a) ; (b) ; P
=
p q
q
(b)
p;
F p q
b ,
(1.2) 1 1
j n
j=l
q
n z_
n!
j=i
The series on the right-hand side of (1.2) is absolutely convergent for all values of z, real and complex, when p ^ q. Further, when p = q+1, the series is convergent if Izl < 1. It converges when z = 1 if
1
Re\
Iv I'. J=i
> 0
J =i
and when Izl = 1 , z * 1, if
> -1.
Re j=l
J=l
If p > q+1, the
series never
converges
except
that
z = 0,
and
the
function is only defined when the series terminates.
In
an
attempt
to
give
a
meaning
to
F
in
the
case
p > q+1
p q
C.S.Meijer in 1941 introduced and studied the special function which is now well-known in the literature as G-function and represented by the following Mellin-Barnes type of contour integral
'
(1.3)
,n z ,q
(a )) p
m,n
(0 )]
p.q
q
where i = V^l
(1.4)
(a) z
1
I,P
O)l,q
J
2ni
*(s)z ds,
y
, z * 0, and
J~\ ro.+s) 7 7 rd-a.-s) *(s) = -i^i p q T~T r(a +s) T T r(i-/3 -s) j=n+i
j=m+i
Here T is the gamma-function and L is some contour in the complex s-plane.
Various
aspects
of
the
theory
and
applications
of
the
Meijer
G-function are available in the books by Y.L.Luke (1969), A.M.Mathai and R.K.Saxena (1973), O.I.Marichev (1983), A.P.Prudnikov, Yu.A.Brychkov and O.I.Marichev (1989). In 1961 C.Fox introduced a more general function which is well-known in the literature as Fox's H-function or the H-function. This function is also defined by the Mellin-Barnes type of contour integral as follows
2
,n
X
,q w
(a , a ) p
p
(0 , b )
r m,n = H. p.q
(a,a)
I,P
X
(0,b)
i,q
q '
q
(1.5) rm, n = H. p.q
where O ^ m ^ q ,
(a , a ) , ,
(a ,a
))
«VV'
(0 ,b )JJ
X
q
$(s)x
2ni
ds
q
O ^ n ^ p ,
7 7 fO.+b.s) T 7 r ( l - a . - a . s ) (1.6)
*(s)
J=
-
1
J=i
p
q
"TT r(a +a s) FT rd-/3 -b s)
j=n+i
and L i s some c o n t o u r ajf j = 1,2,
j=m+i
i n t h e complex s - p l a n e
( s e e C.Fox,
1961).
If
all
,p, and b., j=l,2,...,q are equal to 1, then $(s) (1.6)
is equal to ^(s) (1.4) and Fox's H-function
(1.5) coincides with the
Meijer G-function (1.3).
The
H-function
was
properties are accounted (1963),
A.M.Mathai
and
studied
by
various
in the well-known R.K.Saxena (1973),
mathematicians memoir
and
in
of the
and
its
B.L.J.Braaksma monograph
by
H.M.Srivastava, K.C.Gupta and S.P.Goyal (1982).
The great success and fruitful nature of the theory of hypergeo metric functions in one variable stimulated the study and development of a
corresponding
theory
defined and studied
in
two or
more
systematically the J
variables. four
In
1880
P.Appell
functions F , F , F
J
1 2
and
3
F , which are generalizations of the Gaussian hypergeometric function in 4 two variables. These functions are now popularly known as Appell functions.
Other
hypergeometric
functions of
two variables were
investigated by J.Horn in a long series of papers within a fifty-year period (1883-1939). A list of all these function is given by A.Erdelyi et al. (1953, V.l).
3
The functions
F
to
1
F
and
4
their confluent forms were further
generalized by J.Kampe de Feriet who introduced the function defined by the following series A:B;B' (
(a):(b);(b');
F
x,y (c):(d);(d');
C:D;D' (1.7) A
B
T1 T (a ) ' j=l
j
J
m+n
' ' j=l
C
notation
used
m ' ' j= l
'
j
n
X
D'
m n y
m!
n!
J T <<* > F T <<*'> j
m+n
' ' j=l
on
the
left
j=l
The
j
D
n=0 YJic) 1
B'
F T (b ) F T (b')
j
m ' ' j=l
of
j
(1.7),
J.L.Burchnall and T.W.Chaundy (1940,1941), is
n
essentially more
one used originally by J.Kampe de Feriet (1926).
due
to
compact than the
The
above
double
series is absolutely convergent for all values of x and y, if A+B < C+D+l and A+B' < C+D'+l. Also if A+B = C+D+l and A+B' = C+D'+l, we must have any one of the following sets of conditions (i)
A ^ C, for max{|x|,|y|> < 1;
( ••^ A ^ n
JT
I |1/(A-C)
( n ) A > C, for |x|
| ,1/(A-C) ^ .
+ |y|
< 1.
It is understood (in each situation) that no zeros
appear in
the
denominator of (1.7).
The additional double series
conditions which guarantee
(1.7) when x and y
lie
the convergence of the
in the boundaries of
regions
described by (i) and (ii) above have recently been established by Nguyen Thanh Hai (1990a,c). Readers can
find them in § 14 of the present book.
The Kampe de Feriet function (1.7) of two variables and its particu lar
cases have been studied at great length by a number of subsequent
workers like W.N.Bailey, T.W.Chaundy, J.L.Burchnall and several others. The
hypergeometric
functions
of
one 4
and
more
variables
have
been
discussed
in detail by H.Exton
in his two books
apart from the special cases and
(1976, 1978) where,
important properties, he has given
various computer programs to evaluate certain integrals and a number of interesting applications of these functions in statistics, physics and other fields.
The problems of constructing all distinct triple Gaussian hypergeometric series and establishing their solved
completely
in recent book
regions of
convergence have been
by H.M.Srivastava
and
P.W.Karlsson
(1985). This book also presents various interesting properties for the Kampe de Feriet function (1.7) and its further generalizations.
The study of the functions of two variables has greatly increased after the independent introduction of a generalization of the Kampe de Feriet
function
by
R.P.Agarwal (1965)
functions of Agarwal and
and
B.L.Sharma (1965).
only a difference of notational representation
in them. Here we can
write these functions as follows
(3) ((a(1)^): ((a(2)^ ); ( (a ^)]
0 ,n : m ,n ;m ,n ^
1
2
2
3
3
x,y
(0 < n ): (P (2> ); (P(3))
(1.8) * (s+t)* (s)* (t) x s y Sisdt, (27ri)2
1
L
2
3
L 2
1
where
T7r(l-a: -s-t) (1.9)
j=i
# (s+t) = 1
The
Sharma are essentially the same and there is
J
p
q
i ra-/3u;-s-t TMT r(a(i)'+s+t) F T j=m
j=n +i l
5
l
+i
= G
m
m
*
(1.10)
k
n for k = 2,3.
(T)
rrr(a;k,+T) j=mrr+1ra-^-T
J= n k+ i
Encouraged
k
by usefulness of
variables,
R.S.Pathak (1970),
P.C.Munot
and
the aforementioned K.K.Chaturvedi
S.L.Kalla (1971),
S.L.Bora
functions
of
two
and
A.N.Goyal (1972),
and
S.L.Kalla (1970),
R.K.Saxena (1971), R.U.Verma (1971), M.Shah (1973a,b) and several others have studied function of two variables which are more general than the G-function of two variables given by (1.8). It is interesting to remark that all of these functions are essentially equivalent and differ merely in their notational representations. Some of these authors have named their functions as the H-functions of two variables. Here we write this class of functions as follows: ,
1
2
2
3
3
l
2
2
3
3
p ,q :p ,q ;p ,q *1
(1)
(a
0 ,n :m ,n ;m ,n
(IK
,a
,
(2)
): (a
(2K
,a
,
); (a
(3)
<3KN
,a
)|
x,y
0 ( 1 ) ,b ( 1 ) ):0 ( 2 ) ,b ( 2 ) );0 ( 3 ) .b ( 3 ) )
(1.11) 1
$ (s+t)* (s)<3> (t)x
(27TJ)2 .
I-
1
2
2
y dsdt,
3
I
"l
where n j I T 1-a. j=i
(1.12)
^
J
-a.
(s+t)
}
$ (s+t) = 1
rr rL;i|ta;"(stt)]rir[i-r-b'"(sn)]
j=n +i
^
J
J
6
' j=i
^
J
J
f
m
n
rfr(/3
+b
(k)
]^fr(l-a(k)-a(k)r\
T) }
$ (r) = -^±
(1.13)
•
ri
oA
, k = 2,3.
q
r f r ( a ! k V k ) T ] j - f r(i-3 ( k ) -b ( k M j=nk + i ^
Here as in
—
p
k
IT
fA
i^>>
J
n
J
>J=mk+1
i
(.1.8 J-(1 • 10J all a. , b. J
J
J
are real positive numbers.
J
Later in 1972 P.K.Mittal and K.C.Gupta defined the H-function of two variables which is more general than the H-function in (1.11) and is represented in the following manner: [ H
0 ,ni:m2,n2;m3,n3 nP
rr - n rt n rr l,qrP2,q2,P3,q3
X
'y
I
I (a ;a ,A ) :(c ,y ) ;(e ,E ) ) j j j l. ?i j J L P 2 J J I,P (b ;/3 ,B ) :(d ,5 ) ;(f ,F ) J J j i,q1 j j i,q2 j j i,q3 J
I
(1.14) = ——
(s,t)0 (s)0 (t) x sy \isdt,
Uni)2]
J L 1
where
2
'
3
L 2
n
l
n
7 7 r[l-a r[l-a +a s+A t]
(1.15)
J
J=i
V >t
= "p
l
Ff
r[a.-a.s-A.t] +i
j
j
77
j
m
r[l-b.+0.s+B.t] j
j=i
j
J
n
rf
r ( d -6 s)
(1.16)
* (s) = - ^
rf
—
^2
77 j=m
,
q
*i j=n
J
j
>—
ni-c.+r s) 1
r(l-d.+5.s) 7 7 +i
J
J
j=m
-^—
2
,
r(c.-y.s) +i
J
J
(e ,E ) , j j I,P3
(f ,F ) and J J i,q3
all
a ,A , /3 ,B ,5 ,r J J j j j J
are
real
positive
numbers.
The
standard
work
on
the
theory
and
applications
of
the
last
H-function (1.14) of two variables is the monograph by H.M.Srivastava, K.C.Gupta and S.P.Goyal (1982) where an extensive bibliography of all relevant papers up to 1982 is also contained.
New trend in the theory of hypergeometric functions was initiated by R.G. Buschman who in 1978 defined the most general H-function of two variables as follows
H[x,y; (a,a,A) ; (j3,b,B) ; L g ,L ] (1.17) 9(s, t)x s y~ dsdt, (2TTI) 2
L
s
L. t
where T~Tr(a +a s+A t) 1 _'
(1.18)
j
j
j
e(s,t) = - ^ n "TTr(/3 +b s+B t) J=I
Here coefficients a ,A , b ,B are real J J J J positive, negative or equal to zero.
numbers and they y
may be y
R.G. Buschman has considered various aspects of the last H-function in (1.17) in his several papers (1977-1990).In particular, in the study of convergence of the integral of (1.17) he has found some inaccuracies which were made by other authors. This mistake means that the double Mellin-Barnes type contour integrals in (1.8), (1.11) and (1.14) may be divergent, but at the same time their both inside integrals converge. In
the present
chapter
(§3-4) we
shall
consider
in detail
convergence problem and give its complete solution established
8
the
recently
by the first author. The paragraphs §5 — §9 are devoted to the study of various properties of the Buschman H-function (1.17) in general case.
§2. Definition and notation
Note that Fox's H-function (1.5) can be written in the alternative form as follows
T T r ( a +a s) (2.1)
H[x;(a,a)
1
; 0 , b ) ;L ] m n s
J= i
x
27Ti
ds,
L T T f O / b s) S
where a , b j
It
J= l
are only real, positive or negative numbers. j
allows
us
to generalize this function in the case
of two
variables in the following way.
Definition 2.1. General H-function of two complex variables x and y is called the convergent double Mellin-Barnes integral H[x,y; (a,a,A) ; (0,b,B) ; L ,LJ J m n s t (2.2) 1
9(s,t)x s y
dsdt,
(27Ti)2 . I
- t 1 Js
where
"TTr(a + a (2.3)
e(s,t)
S+A
t)
j=i
TTr(/3 +b s+B t) .
J=I
9
j
j
J
Here m and n are non-negative integers (an empty product is inter preted
as unity);
a., b., A., B. are
a 2 +A 2 * 0, b 2 +B * 0: a , 6 are J J J J J J
real
complex
numbers
such
that
numbers. The variables x and
y are not equal to zero and x
s
= exp{-s[log|x| + iarg(x)]}
and y
= exp{-t[log|y| + iarg(y)]},
in which log|x| and loglyl denote the natural logarithms of |x| and |y|. Also,
the L
and
s
L. are t
infinite
contours
in s-plane and
t-plane,
respectively, such that a. + a s + A.t * 0,-1,-2, ...for s e L , t e L , j = 1,2,...,m. Here suppose that L , L,., a , a , A , 8 , b , B , x and y satisfy some **
s'
t'
j'
j' j
j
j'
j
reciprocal conditions which provide the convergence of the integral in (2.2). These
conditions are
discussed in
detail
in
the
following
paragraphs §3 — §4. Remark 2.1. The H-function contains defined
(2.2) is
very a general object and it
all other G- and H- functions of two variables which were by
various
authors
in
literature
(see,
for
example,
(1.8)
(1.11), (1.14)).
Remark 2.2. The H-function
(2.2) is different from the H-function
which was defined by R.G.Buschman (1978) x
by x and y
by y. It means that x s y
only
by the replacement of
is replaced by xsy
in the
integral (2.2).
We make this replacement in connection with its large convenience in the further
studies and
applications
of
the H-function
(2.2).
This
question is connected with the double Mellin transform of the H-function and it will discussed in detail in §7.
Remark 2.3. Here, as in R.G.Buschman (1978), in
Definition 2.1 we
omit the poles separation supposition for the kernel 9(s,t) (2.3) in the
10
integral
(2.2).
(respectively
This
L.)
known
must
supposition
separate
the
means
poles
that
of
1
the
1
T
contour
L
r (a +a s+A t) j
a >o
j
j
j
(respectively
m ] |" T(a>+a.s+A t)) from J J J A >o
m the poles of ] |" T(a +a s+A t) J j j a
J
j
m (respectively
] f" r(a.+a.s+A.t) under t € L J J J A
(respectively s € L ). s
j
Here
m ] f" a >o
denotes
the
product
over
all
j = l,...,m,
for
which
j
a
> 0, and so on. j
In §3 and §4 we shall prove that this supposition does not influence the convergence of the integral in (2.2).
Till
1977
the H-function
of
the
kind
(2.2) was
introduced
and
studied by R.G.Buschman (1977-1990). He suggested many interesting ideas which are reflected and used in this Chapter. Later in 1983 O.l.Marichev and Vu Kim Tuan discussed various types and notations, applicable to the study of analogous H-function of N variables. By nowadays a great number of original papers of various authors are dedicated to the study of the H-function
(2.2) in general or special
cases. In particular, the theory and applications of H-function
(1.14)
are considered in the book by H.M.Srivastava, K.C.Gupta and S.P.Goyal (1982).
This Chapter presents new results obtained by the first author in the theory of the general H-function (2.2). In particular, the following §3 and
§4 are devoted
to
the complete
problem of the general H-function.
11
solution
of
the
convergence
§3. The convergence region of the general H-function of two variables Now suppose that the contour L c+ioo. Then integral one
variable
in
in the s-plane runs from c-ioo to
(2.1) which is used to define Fox's H-function of
alternative
form
converges
if
(see
A.L.Dixon
and
W.L.Ferrar (1936)) ( m n ^ \ Y. la.I - E lb I • I
|arg(x)| < -j-
(3.1)
Consequently, if the contour L
in the t-plane also runs from C-ico
to C+ico, then condition (3.1) together with the next inequality (3.2)
|arg(y)| < < -\-|\arg(y)|
|B I| E IA I - E IB
is necessary for the convergence of the integral authors (1965),
of
many
works,
R.S.Pathak
for
(1970),
P.C.Munot
and
R.K.Saxena
(1971), R.U.Verma
S.L.Kalla
example, R.P.Agarwal K.K.Chaturvedi
(1970),
S.L.Bora
(1971), M.Shah
in
(2.2). However,
(1965),
B.L.Sharma
and
A.N.Goyal
(1972),
and
S.L.Kalla
(1970),
(1973a,b),
P.K.Mittal
and
K.C.Gupta (1972),have inaccuracies,when considering that (3.1) and (3.2) are sufficient conditions for the convergence of the integral in (2.2). This mistake may be illustrated by the example + ioo ioo +ioo +ic»
f
f rd+s+t)x"ytdsdt.
--ioo i o o -ioo -ioo
Indeed, the last integral satisfies conditions (3.1) and (3.2), but at the same time this integral diverges for all (x,y), x +y
^ 0 . (See also
Exempame 3.1 and 3.2te Thid fact waf roalle( and wa+ mad0 precisa by R.G.Busshman (19783. For convenience se shace call the integral in (2.2) Ry .he cerm H-m78graFo 12
Corollary 3.1. [R.G.Buschman (1978)]. Let such such
that that
m n £ lau+A v| - £ |b u+B vl > K |u| + K |v| j j X y j=i j j=i j
(3.3) for
exitt ■1tt K > 0, K > 0, * y x v
there there
all u,v € R. Then the U-integral aJJ
in (2.2) converges if
|arg(x)| < TTK X /2 , |arg(y)| < 7iKy/2. Later, Vu Kim Tuan (1985,1987) obtained the next result. Corollary 3.2. The H-integral U-integral such that such that u +v = 1 the next next (3.4)
-|-|-
valid valid
(>) (>) in in (3.4) (3.4) is is replaced replaced
symbol symbol
(<) (<) for at least least
a pair pair
in (2.2) will will
H-ineegral H-ineegral
holds holds
( m m n ) ) varg(y)|.. £ |a.u+A v| - £ | b . u + B v | > |uarg(x) + varg(y)| JJ j=i JJ [j=i JJ j=i JJ
If the If the inequalyty inequalyty one one
in (2.2) (2.2) converges converges ifif for for all all u, u, vv ee R, R,
inequalyty inequality
of
two real real
by the
opposite opposite
u and and v, then then
numbers numbers
the
ddverge. ddverge.
In 1990 the first author got the next rather simple criterion for finding of the convergence region of the H-integral in (2.2). Theorem 3.1. 3.1. Let
the
Re(s) and Re(t) are are restricted restricted •
f sy
s-\s-\v\-w r / ) y * / y / ^ r *
*-\ n
Lg and L
contouss contouss
-r\>~ s*ki r ■* sJ *r*sJ
for
s € L
+ J-k *r> -f s-\ 1 1 *~\t
have have
vertical vertical
form, form,
and t € L .Then the
i v\f*
i -r\£**~f* ir*1{-t-i£*c
rav-^o
i.ee,
U-integral H-integral
-P,/-\ l
k 2 .2 2,.co,m+ne k = 1 m
(3.5)
p = k k
m ^^ ) )
, (a I fa det j
^ ''
j= = ll
a
vv
A ] , (arg(x) ] ,, , JJ sgn(m+l/2-j) sgn(m+l/2-j) > >— — det det
A
y y k k k k
a = b ,A = B j m+j m+j j
where
If
we replace replace
(<) for for at least least diverge. diverge.
''
* * ''
for j = 1,2
the inequalyty inequalyty
JJ
symbol symbol
II a a
vv
kk
argiy)]
A ]A]'' kk ''
n.
(>) (>) in (3.5) (3.5) by the opposite opposite
one k, k=l,2,...,m+n, then then
,,
one
the U-integral H-integralin in(2.2) (2.2) wiii wiii
, ,
In order
to prove
this theorem we need
lemma. 13
the following
auxiliary
Lemma 3.1. Let 7 y , a , A € R, j = 1,2,...,m. Then Then for for J J j J (3.6)
m U 2V * * 00, , R, u2+v2 £ r.|a.u+A u,v € R, g(u,v) = J]y | a u + A vv| | > 0, 0, u,v j=i j =i
J
J
.
it is necessary and sufficient
that
m I 'a'a A ]]I| m I AJ J J p= = ) y det j > 0, p 0, k k Z^. J| a A I' L^. J| a A I
(3.7)
In this case the following inequalities
(3.8)
g(u,v) * *
P, — |V| J L_ | , V| , |a
(3.9) (3.9)
g(u,v) g(u,v) £ £
,m. ,m.
k k ''
^ ^ k k
j=i j=i
1,2 kk==1,2
are true
a k * 0, k = 1,2 1,2,....m; a m; k
k'
Pk — — |u| |u| ,, | Ak'
A * 0, 0, k k= = 1,2,...,m. 1,2,...,m. A * k
Proof. Although the function g(u,v) (3.6) is not linear in the plane it is linear in sectors of this plane. Note that all lines L a u + AJv A v = 0>, 0>, Lj = {u,v € R, aju j
j
J
j = 1,2,...,m, j
separate the (u,v)-plane into nonintersecting sectors, which
contain
the origin of co-ordinates as common vertex . It is evident that in each sector
the function g(u,v) is linear with respect to the variables u
and v. Let the point (u ,v ) be located in some sector which is bounded by two neighboring lines o and L. . Then there are points (u ,v ) and 1
by ,v o on the lines L ana L 2 22
j ji
j j2
l
2
.esThenively,
arch
that
( , v +v , 0
1 2 '
v
, u + v Hencl nue to the lLnearity of v(u ,) in hhis sector =u +ave 0 O l 1a 2" v »■-^u = >>u_+ u_+ ——..iiuunnvvjj eeiuiu ,,vv JJll nnis nnis laso last,equal equalvv)y )y ni nitows tows se se to to conclude conclude 2
2
that for g(u,v) v 0, u + v * 0, is is necessary ald sufficient that that for g(u,v) v 0, u+ v * 0, is is necessary ald sufficient that g(u,v) > 0 for all pointv )u.v) located on all lines L^ This is g(u,v) > 0 for all pointv (u.v) located on all lines Ly This is equivalent 1o conditions 23.7). In fact, since the point (A ,-a ) equivalent 1o conditions 23.7). In fact, since the point (A ,-a ) beaongs ro the line L , then *0 ,ait beaongs ro the line L , then *0 ,ait t a t a 14
m k
k
m^°)
j k
j k'
J
To obtain
'a
r—-.
g(A ,-a ) = F y | a A - A a | =
)
/^
y
J
(3.8) we
J
a
J=i
inequality
A
j
det
p
> 0.
A
k
k
transform g(u,v),
for v * 0, as
follows. g(u,v) = |v|g(u/v,l) = |v|g (y), y = u/v. The function
gjCy) = I
y la y + A |
j=i
is continuous and positive, hence
min g (y) = min g (-A /a ) = min k k R * a ^ o 1 a * o k
. |a |
k
Inequality (3.9) is proved by analogy.
k
■
Remark 3.1. If in Lemma 3.1 the strict symbol of inequality (>) is simultaneously replaced by the non-strict one (£) in (3.6) and
(3.7),
then the statement of this Lemma is also true.
Proof of theorem 3.2. We use the following known estimations (see, for example, O.I.Marichev (1983)) r(ot+s) « S . e x p [ - 7TJT72(S)/2] ,
|x
| « H . expUm(s)arg(x) ] ,
where Re(s) is restricted, Im{s)
—> oo and S 1 , S 2 are of lower order
than the exponential one. With the help of for Im(s),
Im{t)
—> oo, and restricted Reis),
these estimations we get, Re{t)
that
TTr(<x +a s+A t ) t
|e(s,t)x"V l =
j=i
-s
x
T1 1T r o +b S+B t j
j
15
j
-t
y
(3.10)
m £ la Im{s)+A
« S .exp 3
j=i
j
n - £ |b Im(s)+B Im(t) I
Im[t)\ j
j=i
j
j
+ Im(s)arg(x) + Im(t)arg(y)
where S
is of lower order than the exponential one.
Now we denote Im{s)
(3.11)
£ |a u+A.v|sgn(m+l/2-j), j
j=i
a
m+j
= b
j
= v and
m n V |a u+A v| - £ |b u+B v|
F(u,v) =
=
where
= u, Im{t)
, A
m+j
= B
J
j
for j = l,2,...,n.
Then for the convergence of the H-integral in (2.2) it is sufficient that F(u,v) - uarg(x) - varg(y) —> +oo 2
2
for u,v € R, u +v
—> oo.
Since F(-u,-v) = F(u,v), then it is equivalent to
(3.12)
for
u,v
g(u,v) = —
€ R,
u +v
F(u,v) - |uarg(x) + varg(y) | -^ +oo
—» oo. In
as
much
as 2
recalling does not alter
Now let
|arg(x)|2 +
g(u,v) > 0 for u +v
this
* 0.
|arg(y)|2 * 0. Then it follows from Lemma 3.1
that for the convergence of the H-integral that for k = 1,2
g(Au,Xv) = |A|g(u,v) 2
in (2.2) it is sufficient
m+n the following two sets of conditions are true
16
m+n
(3.13)
a
V
A j
det
j
a
A
k
m+n
Y
(3.14)
sgn(m+l/2-j)
'argM
> o,
7T
A j
A
a
^
j
arg(x)
arg(y)'
det
k
a det
2
sgn(m+l/2-j) > 0.
arg(y)
j=i
It remains to be proved that (3.14) follows from (3.13). Indeed, we see from (3.5) and (3.13) that p
> 0, k = 1,2,...,m+n. Then from
Lemma
k
3.1 we get the inequality
F(u,v) =
m+n £ |a.u+A.v|sgn(m+l/2-j) > 0, j=i
J
u 2 +v 2 * 0.
J
2
2
In particular, F(arg(x),-arg(y)) > 0 for |arg(x)| + \arg(y)\
* 0, i.e.,
in this case the inequality (3.14) is true. If
|arg(x)|2 + |arg(y)|2 = 0,
follows from Lemma 3.1
arg(x)
i.e.
that the conditions
then
it
(3.12) are equivalent
= arg(y) = 0,
to
(3.5).
Note that
if we replace the inequality symbol
(>) in conditions
(3.13) by the opposite one (<) for at least one k, k = 1,2,...,m+n, then it follows from Lemma 3.1 that
inf g(u,v) = -co.
(3.15)
^2
Since
the
function
asymptotic estimation then H-integral Theorem 3.1.
in
g(u,v)
(3.12)
is
continuous
and
E
in
the
(3.10) is of lower order than the exponential,
(2.2) will diverge. This completes
the proof of
■
Remark 3.2. Since for the convergence of the H-integral
in (2.2)
the conditions (3.5) are sufficient, then inequalities (3.1) and (3.2), 17
as necessary conditions, follow from (3.5). Also, if a.A. = 0 for all j = 1,2,...,m+n,
then
the
general
H-function
(2.2) breaks
into
the
product of two Fox's functions (2.1) and in this case conditions (3.5) have exactly the form of (3.1) and (3.2). Two following examples show that conditions
(3.1) and
(3.2) do not ensure the convergence of the
H-integral in (2.2).
Example 3.1. Let us consider a particular case of (2.2) ^
r(a+s+3t)r(/3+2s+5t)
=
f(y+s+t) Hy+s+t) Then inequalities (3.1) and (3.2) have the next definite form \arg{x)\
<
7rh narg(yua i i/2TT( At )an same tiha erom (next de hait
i
=
pi
66
2
f
d tt
5
1
i
ee
33 JI " rr tt
I d t If '2
P2= P2=
li
'5 l i I
I 2
5
I
f
x
Mi
11
3
I d J 12
2
= -1 < 0, 0,
M l|
5 5
|
K O -
Consequently, the corresponding H-integral diverges for all x,y € C.
Example 3.2. Let us take one more example of (2.2)
e t ) == e (( ss t)
r(a+3s+3t)ros))r(?-t) r(q-f3s+3t)rQs))r(y-t) T(5+2s-2t)
Then
inequalities
(3.1) and
|arg(x)| < 7T, | a r g ( y ) | < n.
p2=
Idet f -1
Hence
3
the
°1I
3
+
(3.2) have
the following
definite
form
At the same time from (3.5) we have
I f 0 -3 M "l I {[ 2 "2 1 I= det 3 det 3
corresponding
3
H-integral
variables x and y.
18
diverges
for
-6 < 0.
all
complex
Consequence 3.1. Let gence
of
the
H-integral
H integral
be defined
in
k = 1,2,...,m+n. Also, trie ine
p
if
m
(2.2) it * 0 for
PR
by is
necessary
at
\
(3.5). Then
least lor
one all
for
that
p
the
conver-
£ 0 for
k
all
k, 1 * k * m+n, t h e n such
x,y € (L,
that
| a r g ( x ) | 2 + || aarrgg((yy))| |2 2** 00.. Indeed, if there is at least one k, 1 s k s m+n, such that
P Pfc R
< 0,
then from (3.11) we obtain inf F(u,v) = -co. R22 If Up p
k
s s oo , , then it follows from Lemma 3.1 that inf F(u,v) < 0. In
this this case case for for |arg(x)| |arg(x)|
+ + |arg(y)| |arg(y)|
* * 0 0 we we have have
2 R R2
inf g(u,v) = inf [TTF(U,V)/2 - |uarg(x) + varg(y)| ] = -co. R2
R2
Therefore, in both cases the equality why the H-integral in (2.2) diverges.
(3.15) holds valid, that is
■
Remark 3.3. After combining all pairs such that
(a
,A j
AA
j 2
j
i
) =
(Aa , jj
i
22
), j ,j = 1,2,...,m+n, we obtain 1 2
F(u,v) =
m+n rr T |a u+A v|sgn(m+l/2-j) = [[ yy |du+Dv|, jj Ji=l = i jj JJ JJ==11 JJ JJ
where r ^ m+n and d D - d D ji k
converges
if
for
all
k j i
* 0 for j * k. Then the H-integral in (2.2)
k=l,2,...,r
the
following
inequalities
are
satisfied
(3.16) (3.16)
fd J - Ii (d pkk = \) det L JJ p k L-r-> II \I d. j=1 1 j= K
D l. ]i i farg(x) D o |i DjJ D >> --|Udet | - det d k II '' 'I (I k v J
argiy)), argiy)). D D
kk
. II ' ' ;;
If in (3.16) the inequality symbol (>) is replaced by the opposite one (<) for at least one k, 1 ^ k ^ r, then the H-integral in (2.2) will diverge. 19
Now let p > o for all k = l,2,...,m+n. 1,2,...,m+n. Then it follows from Theorem k
3.1 that the convergence region of the H-integral [argM.argiy)] -plane is the intersection of all strips |A arg(x)
(3.17)
argiy)| - a arg{y)\
k
< -^- £ - p .. <
k
2
(2.2)
in
the
k = 1,2,...,m+n. k
k
Note that this intersection is a convex polygon containing the origin of co-ordinates as the symmetry centre. The following consequence describes the maximal rhombus which can be situated in this polygon. Consequence 3.2. Let all for
p
k = 1,2,....m+n, 1,2, ....m+n, and
f
P
, defined
k
numbers
p ■ f 1 pk " 1
k ]]
H = min min 1^ — Hx= 7 rL a *o A
(3.18) (3.18)
by (3.5) be positive
H H = = min mm I<
y
x V° 1 'V I
|a
—
[y ..
Then the H-integral in (2.2) converges if
Vthe ° 1sufficient k" J
condition
Then the H-integral in (2.2) converges if holds valid
the sufficient
condition
k
holds valid
v
k
'
ar a rsM g M
(3 19) (3.19)
H x
arg(y) + arg(y) H y
_n_ < JL_ 2 2
Proof. Since p > 0, then from (3.8) 3.8) and (3.9) in Lemma 3.1 we have k
the following inequalities for F(u,v), defined by (3.11) the following inequalities for F(u,v), defined by (3.11) F(u,v) > H |u|, F(u,v) * H x l u | . Hence it follows that
F(u,v) * Hy|v|. F(u.v) * H y | v | .
F(u,v) > £ cH x |u| + (l-e)H |v|, |v|,
0< 0 <e e< < 1. 1.
Then, we get the inequality g(u,v) = --IL. £ _ F(u,v) - |uarg(x) + v a r g ((yy))|| ^^ | U | [ C H XXT T / 2 - | a r g ( x ) | ] + | v | [ ( l - e ) H TT/2 - a r g ( y ) ] . Therefore, the H-integral in (2.2) converges if
20
|arg(y)| < |arg(y)| < (l-e)H TTT/2
|arg(x)|| << eH |arg(x) eH TT/2 TT/2 ,, which is equivalent to (3.19).
Remark halves
of
3.4. the
■
The Buschman' Buschman'ss coefficients coefficients KK sides of
a
rectangle, which
and and KK
can be
in in (3.3) (3.3) are are
situated
in the
honvesgence hegion, seoined by t3.17). Meanwhica, from Consequence the we ver ehat it io possible db se( K
= eM , K = (l-e)H , ons 0 e c < .2 H K H
Remark 3.5. As it was proved in Theorem 3.1, the conditions (3.5) are equivalent to (3.4) in Corollary 3.2. However, unlike (3.4), where 2 2
2 2
this this inequality inequality is is verified verified for for all all u,v u,v ee ( (R R, , u u + + v v 3^j_owi us to verifv ve ifi dnlv roints ,v Now
we
consider
specific
examples
( , for
= =1 1, , Theorem Theorem 3.1 3.1
u + v 2
m+n
illustration
of
the
convergence region of the general H-integral (2.2). Here p and H , H , ° *k x y as above, are defined by (3.5) and (3.18) respectively. Example 3.3. Let
e(s,t) = T(a+s)r(0+t). Then we have p *i
= p
= 1.
2
Hence the convergence region is described by the inequalities |arg(x)| |arg(x)| < TT/2, |arg(y)| < TT/2. Further, since here H = H = 1 , then sufficient x y condition (3.19) brings to the smaller regions |arg(x)| + |arg(y)| < 71/2. Example H
= H
(3.20)
3.4.
Let
9(s,t) = r(a+s+t)r(|3+s-t). r(a+s+t)r(0+s-t).
Then
p
= p
= 2,
= 2 . Consequently, the convergence region (3.17) is described by |arg(x) - arg(y) arg(y)|| << 7T fTT |arg(x) |arg(x) ++ arg(y) arg(y)|| | nT
The sufficient condition (3.19) in this case has the following form
|arg(x)| + |arg(y)) I <
which is equivalent to (3.20). 21
Example
o c 3.5.
T 4- Qe(s,t) r 4.1 = Let
r(l+2s+2t)r(l-3s)r(l-2t) . Then r(l-s-t)r(l/2-s)
from
(3.11) and (3.12) we have ? |u+v| + 2|u| + 2|v| -
|uarg(x) + varg(y)| > 0.
After some calculations we obtain the inequalities
|arg(x) - arg(y)| < 2TT, \arg(y)\
< 3TT/2, |arg(x)| < 3TT/2,
which describe the convergence region. The sufficient condition (3.19) brings to somewhat smaller region
|arg(x)|/3 + |arg(y)|/3 < w/2.
Example 3.6. Let e(s.t) =
r(a.s+t)r(l-3s)r(l-4t)
Then
(3 1 ? )
brings
T(l+2s+3t) to the inequalities
|arg(x)+arg(y)| < n,
\arg{y)\
|arg(x)| < 7T,
< n,
|3arg(x)-2arg(y) | < lire.
The first three inequalities determine the region. Since H
= H
= 1
we again see that condition (3.19) is more restrictive.
Example
3.7.
(See also O.P.Tandon
Srivastava (1986)). Let
G(s,t) =
(1983), R.G.Buschman
and
H.M.
r(a+s-t)ro+s+t)r(y+t)r(-s)r(-t) T(5+s)
The inequalities which describe the convergence region are |arg(x) + arg(y)| < 2n,
|arg(x) - arg(y) | < 2TT,
|arg(x)| < n.
Since H = 4, H = 2 , the rhombus (3.19) is described by x y 22
|arg(x)|/4 + |arg(y)|/2 < <
Exa,™i„ 3.8. r, « TLet *> == Example 6(s,t) o 4 ore:
n/2.
r(a+2s-t)r(^t)r^'+t)r(-s)r(-t) r(a+2s-t)r(/m )r(/T+t)r(-s)r(-t) r(*+s)
XXI
this case we have only two inequalities I—
rtfvl
t o -jy/y f \ 1 I
s
QTT
|arg(x) | < 7r.
Further we have, from H = 3, H = 2, the restricted rhombus x y |arg(x)|/3 + |arg(y)|/2 < w/22 |arg(x)|/3 + |arg(y)|/2 < w/22 Example 3.9. Let e(s,t) = Example 3.9. Let e(s,t) =
Then since here p
= p
the
H-integral
corresponding
|arg(x)| + |arg(y)|
Hcx+s+t)r(/3+s+t)r(-s)r(-t) Hcx+s+t)r(/3+s+t)r(-s)r(-t)
r(r+s)r(5+s) r(r+s)r(5+s)
= 0, then from Consequence 3.1 it follows that diverges
for
all
x,y
e
C,
such
that
* 0. If x,y e IR+, then we have not any information
about convergence or *ivergexce ef tht corwesponding t-integraor
This
qboution is discussed in drtenc io fhe eollowing §4i aboution
§4. The H-function of two real positive variables
As it is showed in §3, the numbers p , k = l,2,...,m+n, defined by (3.5), play an important role in the investigation of
the convergence
of the H-integral in (2.2). In particular, if there exists at least one k,0 ^ k ^ m+n, such that p x,y e C and p
^ 0,
|arg(x)| +
^ 0, then H-integral will diverge for all
|arg(y)|
* 0
(See
Consequence 3.1). If all
then H-inx)gr+l converges provided there are eome additional
conditiont
enf any). It
conditions depena
is not
sifficult
th
underetand
that
thnal
tnly on t , b , A , t , as er Theorem 3.1, bse J J J J L t . Thb presenB paragrapT is mevoted ut cond additidn on o^n /^, L g , L^ ths
rnmnlptp nnmnlpfp
not
conntinn
nf
the complete solution us
nrnhlem thic; n.nhlem
At
lpaqt
ur
qhai s
nhtain
dhp
this problem. At convergence we shall obtain the
criterion, which allow us to determine 23
the convergence or divergence of
the H-integral (2.2) in any situation with respect to a , b,, A , B , a j', 0j', Ls', L. . t First, to illustrate the problem and its solution we consider the following simple example of the general H-function (2.2).
H[x,y; (a,1,0),(0,0,1);(y,1,1); L g ,L t 3 (4.1) r(a+s)T(0+s)
1
x
-s -t, ,. y dsdt.
+
T(r s+t)
(2niV
L, L t s
Here we assume that the contour L
is in the s-plane and runs from
c-ioo to c+ico. Analogously, the contour L
is in the t-plane and runs
from
the well-known
C-ioo
to C+ioo.
Further
note
that
asymptotic
estimation for the gamma-function (A.Erdelyi et al. (1953)) T(u+iv) * V2M |v|u"1/2exp(-7i|v|/2) can be written in the next equivalent form T(u+iv) * \/2i |v+i|u"1/2exp(-7r|v|/2),
(4.2)
where i = "/-f, u,v € (R, |v| —» oo. Hence for x,y € IR+, u = J/n(s), v = Im{t) we have (4.3)
r(a+s)r(0+t) -s -t • r(r+s+t)
X
y
- - | - ( | u | + | v | - | u + v | ) hu+II ^v+II
2
|u+v+i|
E $ (u,v), o where E = const, |uj,|v| 5
= JRe(a)+c-l/2,
and 8
= /te(0)+C-l/2,
8
=
-[Re(y)+c+C-l/2].
Therefore, the integral in (4.1) converges if and only if 24
$ (u,v)dudv < +00. o
(4.4)
or Since
|u| +1 v |>|u+v| for all u,v, then it follows from
(4.3)-(4.4)
that the last integral converges, for instance, if 5 It
is
convergence
evident of
l
< -1,
that
the
5
< -1, 2
the
integral
6
< 0 3
last
condition
(4.1).
Here we
is
sufficient
formulate
the
for
the
following
stronger result.
Corollary 4.1. The integral in (4.1) converges if and only if
(4.5)
5+5 3
< -1, 2
5+5 3
< -1, 1
5+5+5 1 2
< -2. 3
Proof. Note that three lines defined by the equations u = 0, v = 0, u + v = 0, separate the (u,v)-plane into six non-intersecting sectors W ,...,W , where 1
6
w ={(u,v) e \R , u>0, v>0}, W={(u,v) e IR2, u<0, v>0, u+v>0>, 2
W={(u,v) € (R2, u<0, v>0, u+v<0}, 3
W ={(u,v) e IR2, u<0, v<0>, W={(u,v) € R , u>0, v<0, u+v<0>, W={(u,v) e IR2, u>0, v<0, u+v>0}. 6
It is evident that the integral in (4.4) converges if and only if
$ (u,v)dudv < oo o
for
25
k = 1,2,...,6.
In sector W
1
we have
6 6 6 1 2 3 |u+i| |v+i| |u+v+i| dudv.
$ (u,v)dudv o
Below, in Lemma 4.1,we shall prove that the last integral converges if and only if the conditions (u.v)-plane we replace
(4.5) hold valid. Further, if in the
(u+v) by u , (-v) 1
b r e a k s i n t o W . Hence,
by v , then the sector W 2
1
l
$ (u,v)dudv o
| u +v +2 | 1
-7iv e
1
1
1
2
|v - i | 1
5 |u+v+i| 1
1
1
1
|u + 2 |
2
exp
^
6
|v —i| 1
3
1
["-f dVvJ.lvJ-luJ)] dudv
6 |u+i|
3
dudv. l i
Again, using Lemma 4.1, we obtain that the last integral converges if and only if 6 + 5
< -1. Analogously,
$ (u,v)dudv < oo
if and only if
o
6 + 5 2
By the symmetry property of the sectors W
and W
< -1. 3
, k = 1,2,3, with
respect to the origin of co-ordinates it is not difficult to obtain that 26
$ (u,v)dudv < oo o
<=»
$ (u,v)dudv < oo o
for k = 1,2,3. This completes the proof of Corollary 4.1.
■
Now we consider the general case of the H-integral in (2.2). In this case denoting (4.6)
I = {1,2,...,m+n>,
a
= b , A m+j
(4.7)
5.
= [Ke(a. ) + a . c .
j
J
J
f(u,v) = T~Tl a
(4.8)
j
=B m+j
for j = 1 , 2 , . . . , n , j
+ A C - — ]sgn(m+—-j), J
U+
J
J
2
j € I,
2
A v+il j,
j € I,
jel (4.9)
F(u,v) =
V I a.u+A.vlsgn(m+ — -j),
and using the estimation (4.2) for the kernel 0(s,t) (2.3) we obtain the following asymptotic relation, which is more precise than (3.10) |e(s,t)x"sy_t| * H f (J/n(s),/m(t))exp[-F(Jm(s),Im(t))],
(4.10)
where H = const, x,y € IR+, |Jm(s)|, |Jm(t)| —> oo. Therefore, we have
Corollary 4.2. Let
if and only
x,y e (R+. Then
H-integral in
(2.2) converges
if
f (u, v)exp[-F(u, v) ]dudv < +oo
(4.11)
Corollary 4.3. If
are positive, positive
the
then
variables
all
numbers
the H-integral
p , k = 1,2,...,m+n, defined
in
x and y. 27
(2.2) converges
for
by
all
(3.5)
real
In fact, if all p e > 0 such
> 0, then it follows from Lemma 3.1 that there is
that F(u,v) > e(|u| + |v|).
Since f(u,v)
(4.8) is of
lower
order than the exponential one, then Corollary 4.2 allows us to conclude that the H-function in (2.2) converges.
Corollary 4.4. For the convergence necessary
that m ^ 2 and there
1 ^ k ^ m, such
that
det
exists
i j
A J
a
A
k
of the H-integral at least
one pair
= a A - A a j k
j k
in (2.2) it
is
Cj,lO, 1 < j < m,
* 0.
k
Proof. Really, otherwise, if m ^ 1 or m £ 2 such that a A -a A j k
for 1 < j < m, l ^ k ^ m , (Aa ,XA ) we obtain l
k j
then after combining all pairs (a,A ) =
l
n (4.12)
F(u,v) = E |a u+A v|- \
E = const.
|b.u+B.v|,
j =i
Now if there exists at least one j, 1 < j < n such that b A -B a u
J
then F(u,v) —> +oo for u 2 +v 2 —> co. If b.A -B.a then from
j l
* 0,
j l
= 0 for all j, 1 < j < n,
(4.12) we obtain F(A , -a ) = 0 . Further
in accordance with
(4.8) we have sup
exp[-F(u,v)]f(u,v) * f(A ,-a ) = 1.
U,VGlR
Since f(u,v) exp[-F(u,v)] is a continuous function, then the integral in (4.11) diverges, that is why the H-integral in (2.2) diverges too. For example, if in (2.1) 9(s,t) = T(a+s+t), T (a+s+t )T(/3+s+t), 1
T(a+s+t)
r(a+s-2t)
r(/s-s+t)
r(a+s+t)r(0-3s-3t)
r(y-s-t)
then the corresponding H-integrals diverge.
28
r(a+s-t)r(/3-3s+3t)
r(?-s+t)
■
To formulate the stronger criteria we need some auxiliary notions and notations. Here remember that all lines
(4.13)
L
= {(u,v) e R
, au+Av=0>,
j
separate bounded
j
(u,v)-plane by
the
into
parts
jel={l,2,...,m+n>
J
non-intersecting
of
the
rays
of
sectors.
two
lines
Any
sector
which
are
is
called
neighboring lines.
Here we give the exact definition of this notion.
Definition 4.1. Let for j € I 1
if A
= 0 j
(4.14)
-a.sgn(A.) j
if A j
Then two lines L if
a
k
*■ a
and
l
[min(a ,a ), k
and L
all k
1
j
of type (4.13) are called neighboring lines
other
max(a ,a )]
1
* 0 j
a + A
a , j € I, j
or
[-1,
belong
to
the
closed
sets
min(a ,a )] U [max(a ,a ),1] k
1
k
1
simultaneously.
The notion of two neighboring lines geometrically means that these two lines lie one beside another among the
lines L , j € I, in the
(u,v)-plane.
the
j
In
accordance
with
(4.14)
value
a
is the j
projection to
the axis
upper half circle A
=0).
L
= {v=0>, L 3
and L
For
(Ov)
of the point of intersection of L. J and
*
with unit radius (moreover we assume that a. = 1 if
example,
among
= { u+v=0}, two
four
lines
lines L
4
and L 1
2
are not. The following theorem is true
29
L
= {u=0>, are
L
= {u-v=0},
neighboring, but L 1
Theorem 4.1 Let I, 8 be defined
a (4.15)
by (4.6), (4.7) respectively
A j
det
J
a
A j
= -{j € I, det a
j
= 0
k € I;
A
k
(4.17)
k
k € I;
I'. ■ €I\J
(4.18)
k e I;
k
a J
sgn(m+l/2-j),
A
k
(4.16)
and
k
I: ■
j€I
Let
the contour
L
s
run from c-ico to c+ioo, and L, run from C-ioo to t
C+ioo in the complex s- and t-planes
respectively.
Let there also be at least one pair (j,k), j,k e I, such that a A j k A a * 0. Then the H-integral in (2.2) converges for x,y e R+ if and only j k
if
there
with
I),
exists
some subset
I of the set
for which the following
1)
p
> 0 for
k € I.
Z)
p
= 0 for
k e I\I .
3;
p
< -1 for
^k
^k
are true
coincide
simultaneously
0
k e I\I . 0
4J Among lines
L , k e I\I , there k
is
four conditions
may be empty or
o
•k
lines
I (it
(In
the
are no pairs
of
neighboring
o
other
case
additional
condition
8 < -2
required).
In order to prove this theorem we need lemma. 30
the following
auxiliary
Lemma 4.1. Let all E
> 0.
belong
q, Q, d, D, e , E
to (R and e > 0,
Then
j
e
(4.19)
qU Qv
| u + i | d | v + i | D " T T | c u+E v + i | j=i
j
j
dudv < +oo
j
if and only if one of the following four sets of conditions is true 1)
q > 0, Q > 0.
2)
q > 0, Q = 0, D+A < -1.
3;
q = 0, Q > 0, d+A < -1.
4;
q = Q = 0, d+A < -1, D+A < -1, d+D+A < -2. k A = J5.) .
{Here
j=i
Proof.
J
It is not difficult
to note
that
condition
(4.19) is
equivalent to
(4.20)
e-qu-Qv u d y D
-*
5 (c u + £v )
U
Jdudv <
+ro
j J
Now if T < + co, then q ^ 0 and Q ^ 0. Hence we have four various cases with respect to (q,Q): {q > 0, Q > 0}, {q > 0, Q = 0}, {q = 0, Q > 0>, {q = Q = 0>.
First we consider the case q = Q = 0. Note that from the convergence of the two inside integrals in (4.20) it follows that d+A < -1, D+A < -1. Further separating the (u.v)-plane into two domains u ^ v and v ^ u, we have
u v T T ( c u+E v) jdudv j=i
31
u
Since c
J
> 0, E
J
d+A^ du
k
r
v ]~T(e U+E
> 0, then c
j
v/u
>
Jdv
•
^ e + E v/u ^ c + E for u at v £ 1. j
j
j
j
Hence
e
J
, (e +E )
J
I £ (e +E v/u)
J
s maxJ e J ,
(e +E )
J
l
for u £ v £ 1. Now if we denote the simultaneous convergence or diver gence of two integrals by the equivalent symbol =, then the following relations are easily verified
T
l
s
d+A , u du
Consequently, T
U d + A (U D + 1 -I)du,
if D * -1;
u
if D = -1.
D, v dv
log(u)du,
< oo if and only if d+D < -1, d+D+A < -2. Similarly,
8 d D — u v | |(c.u+E.v) dudv < oo
j=i
J
J
v^u^i if and only if D+A < -1, d+D+A < -2.The equality T = T +T
completes the
proof of case 4) of this Lemma.
Now let q = 0, Q > 0. Then, evidently, the condition d+A < -1 is necessary for (4.19). We shall prove its sufficiency. In fact, since c., E
> 0, then for sufficiently large u and v we have
e u ^ e u+E v ^ G E uv. j
j
j
32
j J
Hence 5. j
(e u+E v) j j
r 5. ^ maxi (c u ) j , (e E u v ) \ j J J
6 A j
I . /
Further since d+A < - 1 , Q > 0, then
e
e
-Qv
d D -j^-r 6 j u v | I u J=i
-Qv
d D -Ap . . 5 j , , u v | I (uv) dudv j=i
, , dudv
u
d+A , du
u
e
-Qv
d+A , du
11
1
e
v
D , ^ dv < co
-Qv
v
D+A , ^ dv <
1
From three last relations we obtain CO
00
e
u v
| | (c u+E v ) j
j-i
J
dudv < +oo
J
if d+A < - 1 , Q > 0. The case 3) of this Lemma is proved.
The second
case can proved
by analogy,
but
the
evident. The proof of Lemma 4.1 is thus completed.
Remark
4.1.
If
in
the
integral
(4.19)
any
first
case
is
■
imaginary
part
i
is
replaced by Ai, A e R, A * 0, then the analogous statement also is true. In
this
case
all
these
integrals
are
equivalent
(i.e.
=)
to
the
following integral
e
Proof of Theorem separate
-qu-Qv. ..d. .,D, .iA . , M |u+i| |v+i| |u+v+i| dudv
4.1. Remember
that all
lines L
(4.13),
(u,v)-plane into nonintersecting sectors. The integral
2
on IR
converges if and only if it converges on any sector. 33
j €
I,
(4.11)
Let W be one of these sectors and it is bounded by the parts of the of two two neighboring neighboring lines lines LL rayss of
and LL .. IFor the proof of this Theorem and
k
1
it is sufficient to show that the integral
f(u,v)exp[-F(u,v)]dudv,
(4.21) W
where f(u,v),F(u,v) are defined by (4.8)-(4.9), converges if and only if one of the following four sets of conditions is true
1)
p
2)
p
3)
p
4)
Pk = p,
k
(4.22)
First
suppose
k
*k
that
> 0, p
= 0, p
1
> 0, p
*1
W
> 0.
1
> 0, p
k
= 0, p
M
p
< -1. < -1. < -1, p
coincides
with
< -1, 5 < -2.
the
first
quarter
of
the
(u,v)-plane W = R+ = { (u,v) € (R2,
u > 0, v > 0>.
In this case two neighboring lines L
and L
k
bounding W coincide with
1
the axes (Ov) and (Ou) respectively, i.e.,
(
L
(4.23)
Since L
= { (u,v) e IR ,
±u = 0 >,
{ (u,v) € (R ,
±v = 0 }.
k
and L
are two neighboring lines then all other lines L ,j €
I\{k,l}, lie in the second-fourth quarters of the (u,v)-plane. It allows us
to
conclude
that
for
any
j e I
two
numbers
a.,
A.
belong
(-oo,0] or [0,+oo) simultaneously. Hence for u > 0, v > 0, we have [ | a u+A v| = |a |u+ |A |v j
j
j
j
(4.24) lau+Av+il j
j
« ||a
j
|u+|A
34
j
|v+i|,
to
In accordance with the definition of lines L we obtain |a | = |A | = 1, A = a k
l
k
fa.
A.) A 1
J
det
l
= | A .| |.. J A I
K AJ l a v
k
k
f aa. . v^
k
|A.|sgn(m+l/2-j),
which are defined by J
l 1
P = I P E
J j
j€i i€l
1l J>
1l
p and p
formula formula (4.15) (4.15) as as follows follows p pk = £
JJ
lI aa
J
for j 6 € I. In this case we can write
(4.25)
A. A.)) == la.| la.| j J A A J
JJ
ddet et
J
(4.13) and from (4.23)
j5
= 0 . Hence
1
I a . |sgn(m+l/2-j). |a.|sgn(m+l/2-j).
j€I
J
Further from (4.16) we obtain J = { j e I, A = 0>, 0>, k
j j
By denoting J = I\{J 0
k
1
1
j
and
I\J I\J
k
jj e <=
k
j
U # 0} from (4.17) we have I) J } = { j e 6l , a A *
K' [ V [ ', jj e e
J= { j G J e I, a = 0>.
J UJ J UJ 0 1
O
j
*.- [Z v V [I V V v jJ e€ II \\ JJ
1
Now from r e l a t i o n s (4.8),
1 1
j e€ JJ UUJ J]
0 O k
(4.9) and (4.24) - (4.25) i t follow that
F(u,v) = p u + p v 11
for u > 0, v > 0.
k k
5 5 5 g(u,v) - ~.TTlu+i| j TTlv+il j TTlla.|u+|A.|v+i| jj , j<=J
j€J jGJ k
j€J o o
1 1
J
J
where S = const, u,v —> oo. Consequently,
by applying Remark 4.1 to the case w" = (R+
we obtain
that that
00
00
00 CO 00
r r f(u,v)exp[-F(u,v)]dudv - r r e -Piu-pkv lu+il *\lv+il v2lu+v+i|^33dudv,
0 0 0 0
f(u,v)exp[-F(u,v)]dudv -
0 0 0 0
e
lu+il 35 35
lv+il
lu+v+i| 3dudv,
where
- s-p , w = y 8 = 8-p ,
=y s u
j
jej
M
'
u
*2
j
j€J
k
'k'
*3
^ j€J
1
- 8-p -p Hence .
j
r
M
k
0
=p
r V ^ = pk' "/"a^s = 5-
V» 3
Using Lemma 4.1 to the last integral, we conclude that the integral (4.21) converges if and only if one of four sets (4.22) of conditions holds valid.
Now
let sector
W, bounded
ghboring lines L neighboring
and L
by the parts
of the rays
of
two
be not coincident with the first quarter of
(u,v)-plane. Then denoting
a (4.26)
A
A 1 j
= det
j
j,r e I,
jr
from
the neighborhood
Hence
after
of L
and L
k
some
we obtain a * a l
calculation
we conclude
that
k
,i.e. A l
ki
the following
*0. four
transforms of the (u,v)-plane into (u ,v )-plane (-1)PA (4.27) (-l)P+1a
(-l)q+1a
where (p,q) = (0,0), (0,1), (1,0), (1,1), are linear, and moreover in any case line L., j <= I, is transformed to L. with the following property:
> L s (Ou ),
L < k
k
l
L <
> L = (Ov ).
l
Here among lines L , j = l,2,...,m+n,
l
l
the two lines L , L
j
neighboring. transforms
converts
(u , v ) - p l a n e . l
i
^
k
are also
1
It is not difficult to note that one of the four above the sector
Now from
W
into
sector
W ={u >0,v >0> of the l
1
( 4 . 2 7 ) we have t h a t f o r (u , v ) € W i
l
1
l
a u + A v = (a j
j
=
u Ta A (-1) P +A a ( - l ) p + 1 l
36
+
v [a A ( - l ) q + A a ( - l ) q + 1 l
= u (-1)PA l
where A
,A jk'
+ v (-l)qA , l
jk
jT
are defined by (4.26).
Hence for j € I the line L
*
in
j
ji
(u ,v ) - plane is represented as follows
(4.28)
L* = | ( u ,v ) € IR2, (-1)PA j
\
1
+ (-l)qA
u jk l
i
Since among lines L
v
= oj.
ji l
J
the two lines j
L* = (Ou ) =|(u ,v ) e R2, v = oi k 1 [ 1 1 1 J
and L* = (Ov ) = | ( u ,v ) € [R2, u i
are
i
neighboring
\
i
then
i
all
= ol i
other
;
L.,
j € I\{k,l>
,
lie
in
the
P
second-fourth
quarters
of
(u1 ,v 1 )-plane,
i.e. in
(4.28)
(-1) A jk and
(-1) A. are non-negative or non-positive numbers simultaneously for any j € I. Then for u
> 0, v
> 0, and j € I w e have P
(4.29)
q
|a u + A v | = |u (-1) A j
j
i
Similarly, for u,v
j
j
l
jk
I = |A
|u +|A
ji
jk
+ v (-l)qA
+i\
1
|v . jl
1
> oo
8 |a u+A v+i |
+ v (-l) A
J
= |u (-1)PA 1
jk
1
5 jl
J
(4.30) 5.
I |A. | u , + l A . l v + i | jk
1
Jl
By using the notation (4.26) we can write p , p
(4.31)
ok = EIV-
J
,
(4.15) as follows
".-£%!.
j€I
j€I
From (4.16) it is not difficult to note that
37
j € I.
1
A
=0
«=»
j € J J
jk
Therefore,
and
A
k
d e n o t i n g6 J
= I\{J
o
=0
<=»
j e J . J
j l
U J } = {uj 6 I,A
k
1
jk
A
l
j1
* 0} a s above we
obtain 5
] ~ 7 | |A., |u +IA., | v + i j€l
(4.32)
77
||Alu
1 1
jk
j€J
Now from (4.8),
+|A 1
j
|v+i| jl
(4.9),
7 7
1
|Alv+i|
j'.€ J'
jl
j
1
7 7
j
MA. lu + i |
j[e j ',
jk
1
(4.29) - (4.32) and by using Remark 4.1 we
obtain
T =
f(u,v)exp[-F(u,v)]dudv
FT n v v ' V V 1 J-P ["Z K'V1^1^.1)] J € I
J6I
f -p u-p v k e 1 F1 T1 j€J
-p u-p v e
l
k
d |v + i | 1
k
j
5 77
' ' jej
|u +i | 1
1
j
S 77 '
'
JGJ
I |A
we again see that
|v + i |
j1
1
j
dudv
0
5-p 8-p d-p 1 k |v+i| |u+i| |u+v+i|
By applying Lemma 4.1
| u +|A
j k l
-p
I k
dudv.
the integral
(4.21) on
sector W converges if and only if one of four sets (4.22) of conditions is true. This completes the proof of Theorem 4.1. ■
Now we formulate analogous criteria to establish convergence of the H-integral in (2.2) for complex variables x and y. 38
Corollary 4.5. Let the contour L run from c-ioo to c+ioo, and L,_ run s t from C-ioo to C+ioo in the complex s- and t-planes respectively. Let there also be at least
one pair
Cj,kJ, j,k e I, such that
a Aj k
A.a * 0. Then the U-integral
in (2.2) converges for x,y
j k
if there exists with I),
some subset
I of the set
for which the following
1) | A argix)
- a arg(y)| < up /2
k
k
3) p < -1 for
may be empty or
four conditions for
2) |A arg(x) - a arg(y)I = np /2 1
I (it
'
€ C if
are true
and only coincide
simultaneously
k e I .
for
k e I\I .
k
0
k € I\I . 0
^k
4) Among lines lines
(In
L , k e the
I\I , there
other
case
are no pairs
additional
of
condition
neighboring 8 < -2
is
required).
Geometrically
this
Consequence
means
that
the H-integral (2.2)
converges for x,y G C, if all p > 0 , k e l , and coordinates of point the [(argix), part
of
|A argix) 1
arg(y)] of the [arg(x),arg(y)]-plane lie in the internal the polygon
-
k
formed
a arg(y) I < np /2. k
'
by
the intersection
of
all stripes
If they lie on the boundary
then the
k
condition p < -1 also required. And if it coincides with some vertex of polygon then for convergence of the H-integral
(2.2) we also need that
8 < -2. Finally, we describe process
of establishment
of convergence or
divergence of the H-integral (2.2) for x,y 6 IR+.
Step y
1.
Check m ^ 2 and exist Jyj,k € I such that a A
j k
- A a
j k
* 0.
Otherwise the H-integral (2.2) diverges.
Step
2. By the formula (4.9)and (4.15) calculate F(u,v) and p ,k e I,
respectively. 39
If there exists at least one k, k € I, such that p
< 0, then the
H-integral diverges.
If all p
> 0, then the H-integral converges.
Step
3. Let all p ^ 0. Then find the subset I of set I such that ^ *k o p > 0 for k e I and p = 0 for k € I\I . 'k o ^k m o Step 4. From (4.14) calculate a. for j € I. In accordance with Definition 4.1 find at least one pair (k,l), k,l e
I\I , such that L 0
and L
k
are two neighboring lines.
If not exist, then conditions of the convergence of the H-integral (2.2) holds kind
p
< -1 for k € I\I . k
0
Otherwise, we obtain the conditions p < -1 for k € I\I and 8 < -2, ^k o where 8 is defined by (4.18).
Now we consider typical examples for illustration of this criteria. First we again examine the integral (4.1).
A 1 4.1.
IT i Example
Qr
40 = r(a+s)r(/3+s) ! e(s,t) . r(*+s+t)
Here
(a ,A ) = (1,0),
(a ,A ) = (1,0), (a ,A ) = (1,1). Hence 2
2
3
3
Step
1. Here m = 2 and det
I v
v
Step 2. Step
Pj
3.
= p 2 = 0,
I = {3>, o
8
= m, 1
I\I
o
1
0 '
det
= 1*0. ^0
\-
1
= {1,2}. 8
= <2>,
J
2
r
=
p 3 = 2.
= Re(a)+c-l/2,
J
a
Al ]
= Re(0)+C-l/2, J 3
2
40
= <3>.
8
= -[Re(y)+c+C-l/2].
p
=5+6 2
*1
p
=6+5
^3
Step
Hence L
p
= 5 +5
2
= Re(a-y)-C,
1 3
= Re(a+/3)-c-C-l.
1 2
4. a ^
= Re(|3-y)-c, 3
= 1, a 1
= 0, a 2
= {u = 0>, L 1
= -l//~2. 3
= {v = 0} 2
are two neighboring lines. Consequently,
we obtain the convergence conditions
Re(/3-9-)-c < -1,
Re(a-?)-C < -1,
6 = Re(a+/3-?)-— < -2.
Note that if a = |3 = y = 0, i.e , e(s,t)
r(s)r(s)
B(s,t), where
T(s+t) B(s,t) is beta-function inequality
allows
us
(see A.Erdelyi et al. (1953)), then the last
to
conclude
that
the
corresponding
diverges for all c, C e R.
Example 4.2. 6(s,t)
_ ru+s+t)r(/3+s-t) r(*+s)r(7j+t)
1 Step
ll
1. Here m = 2 and det
=-2*0. 1
Step
2.
p = p = 0 ,
Step
3.
I = {3,4}, o
*i
6
*2
-1
p =Kp = 2
^3
4
I\I = {1,2>. o
= Re(a)+c+C- — 1
5
5
2
= -[Re(*)+c-— ], 3
= Re(/3)+c-C-— ,
2
5
2
4
41
2
= -[Re(T>)+C-— ]. 2
H-integral
J
= {1>, 1
p
=6+6+5 2
*1
3
J
= {3>,
= Re ((3-7-7)) - 2 C + — ,
= 5+6+5
^2
a
= 1, a 3
=0.
= { u+v = 0 } and
l
lines.
C o n s e q u e n t l y , we o b t a i n t h e c o n v e r g e n c e Re(/3-?-T))-2C+—
Example 4 . 3 .
< -1,
Re(a-y-T))+ —
ri
n
0 p = p = 0 ,
I = J
5
3
^4
= {3,4,5,6},
I\I
= {1,2},
J
= J 3
2
5
tfe(y)-C-^-
'l
^6
= {1,2}.
= {3,5},
=p
3
= Re{v)+c+C-
1
+5+5 4
5
— ,
=tfe(/3)-c-— ,
5
= -[/te(y ) - C - — ]. 6
2
1 1
42
= {4,6}. 6
6
= Re(/3+y-/3 - y 6
= J 4
5c = -[Re(R ) - c - - i - ] , =5+5
2
J
5
5
p
^5
0
= Re(a)+c+C-—,
54 =
= - 1 * 0 .
-1
0
1
< -1.
p = p = p = p = 4 .
2
*1
conditions
r(a+s+t)r(y+s+t)r(/3-s)r(y-t) r o i +s)r(* i +t)
9(s,t)
1. Here m = 2 and d e t
J
2
4
= { u - v = 0} a r e n o t n e i g h b o r i n g
Step 3.
= Re(a-r-r?)+ — . 4
1 3
= -l//~2,
= l//~2,
S t e p 2.
= {4>, 4
p
2
Hence two l i n e s L
Step
J
3
2
L
= {2},
4
S t e p 4. a
a
J
2
)-2(c+C).
1
2
Step
4. a
a
= -l//~2.
2
Hence L
L
1
= {u+v = 0}. 2
Consequently, we obtain Jte(0+r-0 -y )-2(c+C) < -1. l
Note that if here 0 = y
l
then provided the last condition holds
we obtain Appell series F . Now we give the Mellin-Barnes type integral representations for 15 out of 34 Horn's functions of
two variables (see
A.Erdelyi et al. (1953)) with the convergence conditions. We write
H(x,y) =
9(s,t)x y dsdt, (2711)"
L
L 2
where t h e c o n t o u r s L
and L 1
1
r u n from c-ioo t o c+ioo, and C-ioo t o C+ioo i n 2
the s- and t- complex planes respectively. We also assume that these contours satisfy the poles separation supposition for the kernel 0(s,t) (in each situation). For
convenience
we
introduce
the
Slater's
notation
(L.J.Slater
(1966), O.I.Marichev (1983)) Hot ). ..T(a ) 1
(4.33)
ro ).. .ro )
8 ,...,0 1
p_
q
i
q
It is to be noted that other functions from Horn's list cannot be represented
by
analogous
integrals
with
vertical
possible only for the other cases of the contours.
43
contours.
It
is
TABLE 4 . 1
e(s,t)
H(x,y)
a - s - t , £ - s , / 3 ' - t , s, t
" a./s.P* 1
F (a.0,0';y;-x,-y)
r
r
»
y-s-t
*
\arg{x) | < 7i, |arg(y) | < n
' 2
r
a - s - t , £ - s , / 3 ' - t , s, t
*,?,&' F2(a,{3,/r;y,y';-x, -y) y,y'
r
'
y-s, y ' - t
|arg(x) | < 7T, | a r g ( y ) | < TT, | a r g ( x ) - a r g ( y ) | < TT
a , a ' ,/3,/T" 3
r
f* a - s , / 3 - s , a ' - t , / 3 ' - t , s, t F ( a , a ' ; 0 , £ ' ;y;--x,-y)
r
*
\arg(x)\
" a,|3 1 4
r
< TT, \arg{y)\
a-s-t,/3-s-t,s,t 1 F(a,P;y,y';-x,-y)
y>y'J
'
y-s- t
r
4
»
y-s, y ' - t
x,y € [R+, 2c+2C-/te(y+y' ) + l < 0
' a-2s-t,/3-t,s, t ]
" «,£' 5
#3(a,|3;y;-x,-y)
r
r
*.
»
y-s- t
| a r g ( y ) | < TT, | a r g ( x ) |
| a r g ( x ) - 2 a r g ( y ) | < TT,
1a r g ( x ) - a r g ( y ) | 44
<
2TT
< 3TT/2,
" a,0 6
r
-.
" //
.
y,<5
4
r
(a,0;r,5;-x,-y)
a-2s-t,0-s,s,t y-s,
s-t
-, J
|arg(y)|
<
n/2,
|arg(x)-arg(y)|
" a,01 7
r
*J
' a-s-t,0-s,s,t $
r
(a,0;y;-x,-y)
r
' 0,0'1
»
;y;-x,-y)
r
V
r
' &1 _7J
< TT/2, | a r g ( y ) |
*
y-s-t |arg(x)|
y € R+f
" a,0 10
r
. 7,7'
1
" 0-s,s,t "
r
3(0;y;-x,-y)-
1
< TT/2,
c-/*e(y) + l < 0
" a-s-t,0-s,s,t
U1(a>0;r,7,;-x>-y)
r
y-s,
y'-t
|arg(y)|
" »
< ir/2,
| a r g ( x ) - - a r g ( y ) | < 7t/2
45
< n/2
y-s-t
|arg(x)|
9
< 7i, \arg(y)\
" 0-s,0'-t,s,t <M0,fr
"
y-s-t |arg(x)|
8
< TT/2
< TT/2
r 11
_
OL
* (a,y>y';-x,-y)
r
r
y,y'J
a-s-t,s,t y'-t *
y-s,
x,y € R+, 2c+2C-/te(y+y')+l < 0
a - s , / 3 - s , a ' - t , s, t
a, a' ,/3 12
2 ( a , a ' ,/3;y;-x,-y)
r
7
r
\argM
H (a, /3; y; -x, -y)
r
| < irf
a-s,/3-s,s,t
CL,(3
13
r
7
'
y-s-t
|arg(y) | < TT/2
-i
y-s-t
-1 | a r g ( x ) | < 7T,
y € IR+,
c+2C-/te(y)+l < 0
' a-2s-t,s,t 1
a 14
r
H (a;y;-x,-y)
r
6
7
y-s-t
*
\arg{x) | < 7i, \arg(y) \ < n/2 \arg(x)-2arg(y)|
1" a - 2 s - t , s, t
a 15
r
. *>6.
< n
H (a;y,5;-x,-y)
r
y-s,5-t
>
x,y € R+, 2c+2C-Jte(y+5)+l < 0
46
§5. Simple contiguous relations for the H-function of two variables In this section we establish
certain
constant coefficients for the H-function
contiguous
relations with
(2.2). Here we simply write
H(x,y) if all the parameters of the H-function of two variables are as in
(2.2), H[a +1] for the contiguous H-function
which
a
is
replaced
unchanged. Similarly
by
a+1,
but
all
other
of two variables in parameters
are
left
we introduce notations H[a±l], k = 1,2,. 1,2,..,m, . ,m, and k
H[p.±l], j = 1,2,..., n. Equalities involving H(x,y), H[a±l],
withJ constant c o e f f i c i e n t s
are c a l l e d contiguous r e l a t i o nks .
H[0.±1],
But
j those
with constant coefficients are called contiguous relations. But those contiguous relations involving only H(x,y), H[a +1], H[/3 -1] are called simple.
The
general
theorem
of
contiguous
relations
given
in
this
section was obtained by R.G.Buschman (1990) and here we little intensify it Now in the same way as for the H-function (2.2), we write e[ct e[ak+l], +1], k = l,2,...,m, and e[|3-l], j = l,2,...,n, for the contiguous function 6(s,t)
(2.3)
in
j
which
a
is
replaced
by
a+1
k
respectively.
and
0 (3
k
by
j
Applying the equalities r ( ll++cat + a s+ A t ) = (a + a s+ A t ) T ( a +a s+A t ) , k
■"
k
k
„
1
_
rr(/s ( 0 -i+b -1+b S s+B + B t) j j
j j
k
k
k k
k
k
0 -1+b s+B t J
J
J
r(p r(0 +b +b s+B s+B t) t)
j j
j j
Jj
j j
we we obtain obtain m m
nn
ii- m m
nn
-. -.
y a eta +i] + y c.et^.-i] = y ^kak + y €.0. e(S,t) y a e[a +i] + y k k= = ll
(5.1) (5.1)
rrm + r[ r m ) \ \ + [ ) \ \ L v k= l L I k= 1
for any 7)
'k
, k = 1,2
^kak + y
•k= l l "- k=
j= j= l l
+ +
y
€.e[/3.-i] =
j=l j=l
^ ^ . e(s,t) -I -I
n ^ ( m n ^i )n € > . ^s + ( \m 7)kAk + n \ C.B.^ t i e(s,t), ) €>. s + \ 7>kAk + \ C.B. t e(s,t), j= l > ^ k= l j= l ' J j= 1 J V k= 1 j= 1 ) J m and ? , j = 1,2 ^Jj
47
n.
0-1, j
Consequently, this allows us to formulate the following theorem.
Theorem 5.1. Let
m+n numbers
T? , k = 1,2, . . , m and £ , j = 1,2,. . . , n, k
satisfy
the next two
j
equations > i)k k a + ) £ b = 0,
f
\ L
L J J
k=l
j=l
(5.2) m
n
> 7) A k=l
Then the following
simply
C B
= 0.
j=l
contiguous
m
(5.3)
+>
relation
n
is
i- m
n
^T, k H[a k+ l] + £ W - l ] = £\«k k=l
true:
+
E ?A
H(x,y).
j=l
j=l
This theorem is simple to prove. Here it is sufficient to note that T) , £ if constant coefficients 7)
satisfy ((5.2), then the second addend in
the right part of (5.1) is equal to zero.
Theorem 5.1. shows that all simply contiguous relations are defined by m+n roots (T) , 7) ) of system (5.2) of two equations with respect to the known a ,A ,b ,B . Evidently, the rank of this system is equal to 2 y H "
k' k' j* j
(otherwise
the H-integral
diverges).
relations for the H-function
Hence
there
exist
m+n-2
basic
(2.2) of two variables, which define all
other simple contiguous relations. We describe it by the following two examples. Example 5.1. e(s,t) =
Hex +s+2t)T(a -/~3s+t)r(a +s) .
r(^-t) Here in accordance with p
= 2+/~3,
p
=2,
p
(4.15) in Theorem 4.1 we have p
=3.
Hence
converges at least for x,y € IR+.
48
the
corresponding
= 2+2vr~3 H-integral
Further the system (5.2) has definite form
7) -V~3 7} + 7) = 0, 1
(5.4)
2
3
27) + T) - £ 1
2
=0.
[T) , TJ ,-TJ +Vr1^7} , 2TJ +TJ ]
It is not difficult to see that 7)
^1
for
any
7) ,
is the root of the last system of equations with respect to four
unknowns
[T? ,TJ ,TJ ,£ ]. Hence
from
(5.3)
we
obtain
general
simple
contiguous relation
77 H[a
+1]
+ 7) H[a
+1]
+
(/~3T) -TJ )H[a
+1]
+
(2T? +7) )H[0
-1]
[a T? + a i) + a (V~3T) -TJ ) + 0 (2TJ +TJ )]H(x,y). ,J 11 2 2 3 2 1 1 1 2
Evidently, (T) ,7) )-space
two and
vectors
they
[O^.v'-3,1]
[1,0,-1,2]
and
contiguous
relations
(1,0)
define
for
and
two
(0,1)
following
respectively. this
are
roots
the of
Consequently,
H-function
can
basis
system all
simple
by
linear
obtained
combinations from two basis relations
H[a +1]-H[a +1] + 2H[0 +1] = (a -a +2/3 )H(x,y),
H[a +1] + /~3H[a +1] + H[0 -1] = (a +/l?a +0 )H(x,y). 2
Example 5 . 2 .
3
G(s,t) =
1
2
3
1
T(a + / ~ 2 s + t ) r ( a - s + t ) T ( a - s ) T ( a - t ) —
r o -s-t)r(js -s)
49
of
(5.4)
= 3V~2,
Here after some calculations we obtain p
p
= V~2,
p
= 2,
'l *2 ^3 p = V~~2,p = 2+V~^2,p = 2. Hence the corresponding H-integral converges
4
5
6
at least for x,y e IR+. From (5.2) we obtain the next system
( V ^ T ) -1) -n -£ -£ = o, 1 I V
With
respect ^
to
six
2
3 ^1 ^2
TJ +TJ -7) " 5 = 0 . 1 2 4 ^1
unknowns
[TI ,TI ,TJ ,TJ ,6 ,6 ] we V 2 3 4 ^1 ^2
If) ,V >T) >T) >i) +T) ~V , (/^-1)TI -2T) -7) +7) ]
for
any
have
the
17 ,7) ,7) ,7) .
root Hence
V 2 3 '4 1 2 4' 1 2 '3 4 * V 2 '3 4 from (5.3) we obtain the following general contiguous relation
77 H[a +1] + 7) H[a +1] + 7? H[a +1] + 17 H[a +1] +
(^^-irjjHt^-l] +
[(Vr^-l)7)i-27)2-773+774]H[^2-l]
= {a 7) +a T) +a 7) +a T) + B (TJ +T) -T) ) 1 1 2 2 3 '3 4 '4 1 1 2 4
+ ^2[(/^-l)77i-2772-7)3+774]>H(x,y).
If vector (7) ,7) ,TJ ,7) ) is equal to (1,0,0,0), (0,1,0,0), (0,0,0,1),
then
we
obtain
four
following
basis
simple
(0,0,1,0), contiguous
relations respectively Htc^ + 1] + Ht^-1] + ( T T 2 - 1 ) H [ 3 2 - 1 ] = [e^+0 +(iT2-l)0 ]H[x,y],
H[a2+1] + H[^-1]-2H[02-1] = ( a ^ - 2 0 )H[x,y],
H[a 3 +1]-H0 2 -1] = (a3-^2)H[x,y],
H[a +1] - H[p -1] + H[0 -1] = (a -0 +0 )H[x,y]. 4
1
2
4
50
1 2
§6. Main properties for the H-function In this section we assume that the H-integral in (2.2) converges. By Hartogs
theorem
H-function
it
is not
difficult
to obtain
that
the general
(2.2) is analytic at least on the region described as
follows: r
m
n
|arg(x)| < \ u
jj==i i
j=i
m
n
j =i
j=i
|arg(y)| < L
Here, as usually, we denote (a,a,A) = (a ,a ,A ),.
(a ,a ,A ), m m m
(0,b,B)n = ( f ^ . b ^ ) , ,
O
i
m
l
l
n
,b ,B ). n n
It is not difficult to get two evident properties H [x,y; (a,a,A) m-l
, (/3 ,b ,B ); 0,b,B) ; L ,L.] n
n
n
n
S
t
H [x.y; Coe.a.A)^; O . b . B ) ^ ; L ^ ] .
|H [x,y; (a,a,A)m,(0,b,B)n; L s,LJ| t J dxdy ^ = -^- H [x,y; (a,a,A) , (0,-1,0), (0,0,-1); xy m (0,b,B) ,(1,-1,0), (1,0,-1); L ,LJ. n s t Further we will consider the case of the H-function where the contours L and L^ are vertical lines L and L_ with real parts c and C s t c C respectively, i.e. 51
L
= {s, Re(s) = c>, L c = {t, Re(t) = C>.
Suppose that the corresponding H-integral H[x,y; (a,a,A) ; O,b,B) ; L Lc,L ,Lc]] (6.1) = -- i i
— (27ri)2J
L
J J
r
C
— (27ii)2 JJ
J J
t 9(s eCs,Dx'V^sdt =— —)t)x"V dsdt =
Re(t)=C
L
e(s,t)x"ssy"ttdsdt
Re(s) =c
converges. By suitable replacements of s and t in the last integral it is not difficult to obtain that 1. The change property xpyqH[x,y; (a,a,A) ;(0,b,B) ;(|3,b,B) m
n
; L ,Ln] c C/
(6.2) = H[x,y; (a+pa+qA,a,A) ;(0+pb+qB,b,B) ;(0+pb+qB,b,B) ; ; L_ _ ,, L ]; ]; 2. The strain property H[x p ,y q ; (a,a,A)m; (0,b,B)n; 1-^,1^] (6.3) = =
X T^H[X'y;
(a>a/P>A/<*V^b/P>B/q)n > Lc 'LCq ]; (a>a/P>A/^m^^b/P>B/q)n ;L C pP,LCq];
3. The linear property H[x,y; (a,a,A) ;(0,b,B) ; L ,L ] (6.4) k p h = I| A | H [x y ,x q y ;U,ka+ PrA,qa+hA) .;(/3,kb+ ;L ,,L [xV,xV'.Ca,ka+pA,qa+hA) ;L LCnn ], ], (/3,kb+pB,qb+hB) I J J -i J P-B,qb+hB) - I n c m
J
II
where A = kh-pq * 0
J
c
r
-i
m
r-
i
= (hc-qC)/A, C = (-pc+kC)/A.
n
c
C
Here in a l l the
above formulas D a k h are real constants , and pq * 0. Now
we
p^tabli^h
a^vmntot i p
p«?t imat i nnc;
for -far
thf
H-fnnnMnn
(6.1). (6 1) From Definition 2.1 2 1 we have (6.5)
|x" |x"ss|| = |exp|-sriog|x|+iar^(x)l||= |expj-s[~log|x|+iar£(x)lj| = eexp[arg(x)7^(s)j|x|' x p f a r g ( x ) 7 f l 2 ( s ) j | x | ' CC. .
52
Similarly, |yt_t||== exp[arg(y)I/n(t)l exp[arg(y)I/n(t)l |y|" |y|"CC.. |y
(6.6)
Since the H-integral
in (6.1) converges, then as is known from
§3 — § 4 we have
((6.7) 6.7)
M = JJ_ f \ f|e(s,t)|exp[ar*(x)I«(s) L(s,t)|exp[ar*(x)I«(s) + arg(y)Imit)1 |dsdt | < +<*, 4TI 2 J 4TT L
J
L
J
CLc
Denoting the H-function (6.1) by simple symbol H(x,y) from (6.5)-(6.7) we obtain |H(x,y)| * M |x|'C|y|"C for x,y € C.
(6.8)
In accordance with Definition 2.1, the contours L and L„ do not c C intersect any singularities of the integrand e(s,t), i.e. intersect any singularities of the integrand e(s,t), i.e. ReU + a c + A C) * 0, -1, -2, ... .. . j
for j = 1,2
j j
Jj
m. Hence for all j = 1,2
m there exists e > 0 such
that /te(a.+a s + A.t) * 0, -1, -2,... for s,t, such e+c and -e+C < Re it) function
9(s,t)
that -e+c < Re{s) ite(s) <
< e+C. This means that for these s and t the
(2.3) defined
by
the ratio
of
two products of
gamma-functions is analytic. Consequently it is not difficult to obtain that H(x,y) = —
9(s,t)x' e(s,t)x'ssyy_tclsdt, \isdt,
2
(27ri) J J L
C Lc l
l
where (6.9)
-e+c -e+c < c < c e +c, +c, l
-e+C -e+C < C < e+C. l
Applying the estimation (6.8) we have -c -l |H(x,y)| * M|x| 1|l| |l| .
53
for any c , C , satisfying the conditions (6.9). The last estimation can be written symbolically as follows |H(x,y)| *M|x|- C ± G |y|- C ± e
(6.10)
where e is a sufficiently small real positive numbers.
§7. Let
The double Mellin transform
the function f(x,y) be defined
in R? =
(0,+oo)x(0,+a>). Then (0,+oo)x(0,+co).
classical double Mellin transform of function f(x,y) in point (s,t) € C cl ssiced thu lelMowlng inansfor 00
mjf(*.y);s,t^
(7.1) (7.1)
00
p p p
f(x,y)xss 1yyt f(x,y)x
= f*(s,t) f*(s,t) = = =
X
dxdy. *dxdy.
0 0
This transform plays important role in the construction of the theory of integral transforms and convolutions. Here we give main properties of this transform. Theorem Theorem
Let Let
7.1. 7.1.
IIIL cgla.1
III
L lie
nitcgia.1 that that (7.2)
in
Liic r ±gtii c
r lgtl
aloo
the
L
pixl
L
parL
< Res))
l
l
Let
function function function
l
function
continusus continuous continusus COilVSlgcS
I OL
d.1 1
au£>oiutciy
converges
ior
a±± s,t,
w .1 ) au£>U ILL t c ly
w .1)
< E
< Reit)
on on
R+ and R+ and
s,t,
the the
SUCH
such
< E E.
2 2
2
f(x,y) be reprenented
f(x,y) = — l — —
(7.3)
c
and
l
2
f(x,y) be f(x.y) be
by the
ffrmula
F(s,t)x"sy" F(s,t)x"sy_ttdsdt,
2 2
(27ii) (2ni) J\ J L
where
En L^ L
respeceively respeceively respeceively
and
L
c
mean
(heee (heee
f*(s,t) = F(s,t) for
e e
vercical vercical
< < c c < < E E , , e e all
s,t such
C
lnnes
L
c wihh
<
54
Reis) Re{s)
Then Then
real
parss
C
and
3Jt 3Jt
= c, Rei))
= C.
c
Theorem 7.2. Let theee
srrsps
the
1. Function Function Function
next
s,t belong
to
conditions it ions
hold
strips
defined
by
(7.2) and
in
s,t beoong
to
F(s,t) F(s,t) is is analytic. analytic. analytic.
2. The following 2.
integral
converges
(7.4)
|F(s,t)|dsdt << +O0 +«, L
C Lc l
for
all
(7.5)
c , C , such l
l
t that hat
l
e < c < E ,, I i
ll
from
strips
(7.3) it it
e
ll
2
3. |F(s,t)| 3. |F(s,t)| -> 0 for for
Then
the
2
11
2
||s|, s | , |t| | t | -> -+ --Hoo. Hoo.
foiiows follows
that
f*(s,t) f (s,t) = F(s,t), where
(7.2).
Readers interested in the proof of these theorems can find them in I.S.Reed (1944). Here we note that integral (7.1) absolutely converges. In fact, since function F(s,t) is analytic in the strips (7.2) then from (7.3), (7.4) we obtain f(x,y) f(x,y) = — — F(s,t)x"sy"tdsdt, (27Ti)2 J J JJ
(7.6)
L
C
for
L
ll
c
c , C 1
|| ff (( x x ,, y y )) ||
1
s a t i s f y i n g (7.5). Therefore -C -- cc -u 1 < < M Mx x * ' y y 1 where where
M= J J_
ll
for
all
|F(s,t)dsdt|
4TT 2 J J L
C
JJ L
ll
c
ll
55
x > 0,
<+oo. <+c».
y > 0
we h a v e
Hence for sufficiently small e we obtain -c -c -C -e |f(x,y)| < M x 1 y 1
for x > 1, y > 1,
-c +e -C -e |f(x,y)| < M x 1 y *
for x < 1, y > 1,
-c -c -C +e |f(x,y)| < M x * y *
for x > 1, y < 1,
-c +e -C +e |f(x,y)| < M x l y l
for x < 1, y < 1.
—
The last inequalities allow us to conclude that f(x,y)x R6(s)-1yR6(t) 1 € L(R+), i.e. the integral in (7.1) absolutely converges. Theorem 7.3. Let 9(s,t), defined
by (2.3), satisfy
the
following
conditions Re(a +a c + A C) * 0, -1, -2, ... j
(7.7)
j
|e(s,t) dsdtl < +oo. L
Then for
j
the H-function
C Lc
H(x,y)
e(s,t)x"sy"tdsdt
H(x,y) = 2
(2Tui)
L
we obtain
the next
C Lc
equality
3JI JH(x,y); s.tj = 9(s,t) for
|/te(s)-c| < e,
positive
|Re(t)-C| < e, where
e
is
a
sufficiently
small
number.
Proof. Since Re{a
+ a c + A.C) * 0, -1, -2, ..., then there exists 56
Re{a
+ a s + A t) * 0, -1, -2, ... , j
j
j
(7.8) for Consequently
the
|/te(s)-c| < e , |/te(t)-C| < e .
function
6(s,t)
(2.3)
is
analytic
in
the
strips
described by (7.8).
Further from (7.8) and Theorem 4.1 it is not difficult to observe that
|0(s,t) dsdt| < +oo, L
C
L
l
c
l
where |c-c | < e , | C-C | < e , e
is a sufficiently small positive num
ber.
applying
If
set e = min(e ,e )
and
Theorem 7.2, we obtain the
Theorem 7.3. ■
§8. Series representations for the H-function of two variables
In a general case the problem of representing the general H-function of two variables in terms of the residue sum in the integrand function poles, which is very important for the theory and applications, involves great difficulties, which cannot be overcome yet. An attempt to use the multidimensional residue theory, given in an account of L.A.Aizenberg and A.P.Yuzakov (1979), A.K.Cih (1988),
to the
double
It
H-integral
(2.2) has
not
given
an
essential
result.
is
connected with complicated behaviour of the integrand function 0(s,t) (2.3), when s,t —> oo, s,t € C.
However, in a great number of particular cases the H-function (2.2) can be represented by the sum of double power series of hypergeometric 57
type. In this section we shall show that if the integrand function 9(s,t) (2.3) has the next form
m
m
o
2
6(s,t) = T~Tr(a -a s-A t)rTr(a'+a's)rTr(a ,, +A't) > : J J J j=i' J J j=i' j j j=i
(8.1)
where
m
1
a
£ 0, A £ 0
j
for
j=l,...,m;
j
a' > 0,
o
j=l,...,m;
j
^
A' > 0,
l
j
j = l,...,m , then the corresponding H-function can be represented by the sum
of several
double
Theorem 8.1 below).
series
Here
under
we note
some
that
additional
the situation
conditions
(see
a. ^ 0, A. ^ 0,
a' < 0, A' < 0 is leaded to (8.1) by the replacement x by x
and y by
y"1 in the H-integral (2.2).
Here we also present several examples, which give solution of this problem in the other situations (unlike 8.1). The method
to establish series representations of the H-function
used twice in Slater's theorem, which is based on the residue theory of the
gamma-function.
This
well-known
theorem
appeared
in
L.J.Slater
(1966) and later it considered in detail in O.I.Marichev (1983).
Now denote
(a,a,A) m
= (a ,a ,A ),...,(a ,a ,A ). i
l
l
(0,b,B) n = ^ . b ,B ),
where a , 0
m -
m
m
(/3_,b_,B_), n
n
n
<=C, a , A , b , B € IR such that 8 +b p+B q * 0,-1-2, . . .
for all j = 1,2,...,n; p,q = 0,1,2,.. type series
58
Then the double hypergeometric
in
^ (8.2)
F [x,y; (a,a,A) ; (0,b,B) ] = m n
1 Tr(a +a p+A q)
xPF y 4
^ 0 TTrOj^p^q)
p! q!
> /
_ -^-1 n
j j
q
j
P - - j-l
converges absolutely for x,y € C, if
m
(8.3)
n
m X = 2
X = V a.- V b -1 < 0, j=i
j=i
/ A" / j=i
n B _1 <
°'
j=i
If A > 0, A > 0, then series (8.2) diverges whenever x * 0 and y * 0. 1
^
2
This fact follows immediately from Horn's method for double series (see J.Horn (1889, 1931)) and the asymptotic behaviour of the gamma-function. If all these coefficients a , b , A , B j
j
j
are equal to 0 or 1, then the j
corresponding series is different from Kampe de Feriet series only by some constant. This known series will be studied in §14.
Now we consider the following particular case of the H-function (2.2) in the next form
H[x,y; (a,-a,-A) , (a', a' , 0) , ( a " , 0, A' ) ; (/3,b,B) ; L , L. ] m m m n s t o 1 2 (8.4) m
(2iri)2
m m o 1 2 Y~\ r(a.-a -s-A.t)"T~[ r(a'+a's)"TT Ha'+A't) } r J J j j J J j=l j=l -s -t . .. n_ J = l J x y dsdt. _ n ^ b B ) Ll L V t j t ~s j=l
59
For (8.4) there is the next theorem.
Theorem 8.1. Let
(a
£ 0, A
£ 0
for j = 1,...,m ;
(8.5) a' > 0
Let all be
left
simple,
poles i.e.,
simultaneously
(8.6)
a
(8.7)
j
Let
i
J
s = -p and *
1
poles
the contour
+ a 2
finally,
in the right
of equality
j
s = -q, M 2
a'.'+ A', t = -q, J
2
function
2
respect
(respectively
L
f
m
part
of (8.4)
cannot be
satisfied
j=i
j
1 2
2
part
of
1
2
1
j = 1,
m 2
(8.4) satisfy
the contours all
L
the
and L ,
the poles
(respectively
of conditions
j=i
m 2
o
n
j=i
j=i
j=i
60
J J,J = l,...,m ;
i
j * j •
m when t € L
two sets
J
of
T(a' '4-A't), j = 1, . . . ,m ,) from
(8.8) m
2
to
m
j=i
i
L ) must separate
(respectively
the following
j * JJ ;
J
in the right with
of f(a -a s-A t), j = l
Let,
function
two sets
j
supposition
T(a' +a' s), j = 1, . . . ,m poles
a
J
i
the integrand
separation
and A' > 0 for j = 1, ,m
p,q = 0,1,2,...
a " + A' t = -p and J
i.e.,
j
of the integrand the following
for
+ a 1
for j = 1, . . . , m
be also
s € L ) .
valid
the
m o
m
a
1
+
E j E a j "Z j=i
j=i
m o
|b| > |arg(x)|
j *
j=i
m
1
A +
E i Z AJ " Z J=I
J=I
|B| >
J « i a r g ( y ) i,
J=I
(8.9) m +n o
m
(a det
j
sgn(m +1/2-J) + A \ a' + a \
j
A
A'
A k
j=i
m
A 1 j=l
k
j=l
f arg(x) arg(y) ) det
a k
a
m +r 0
Then the H-function series
, for
A
all
k
= b , A r
m +r 0
= B , r = 1,...,n. r
(8.4) can be represented
of form (8.3) as
k = 1 , . . . , m +n, o
by the sum of
double
follows
H[x,y; (a,-a,-A) ,(a',a',0) ,(a",0,A') ; (0,b,B) ; L ,L.] m m m n s t 0
m
1
2
m 1
a'/a' a'VA'
2
k
k
■II-
h
h
y
1/a'
„
x
F
k
1/A' h , -y
(8.10) a a ^ k ^ h . a A a+ — a + A ,— , — a' A' a' A' k
h
a
h
a
A
» n
k
k , a a - — a ,- —
m -l l
h
a
A A
a
,
_
0- Jib --*-B ,-*- ,- *.
- — A ,0,- — A' A' h
_ ,0
a' k
h
61
A' h
a' k
A' h
where k , a a - — a ,- —
_ , 0 m -l 1
a
1
>
a - — 1
a
n
a ,- —
,
,0
k-l
1
a
k
a k
a
a a k k-l - — a ,, 0 , k-l a a
a k
(8.11)
k
k
a a k , k+l ,0 - — a ,k+l , k+l a a k
x
k
a k - — a ,m , m l a l k
m 1
, a k
and so on.
Proof. At first we note that according to Theorem 3.1
conditions
(8.9) are required for the convergence of the double integral in the right part of
(8.4).
But
the two
inequalities
in
(8.8) provide
the
absolute convergence of all the double series which are involved in the right part
of
(8.10)
(see also
(8.2) and
(8.3)).
Hence
the double
integral in (8.4) can be evaluated as follows. Transform integral (8.4) in the next form
2
1 27i i
TTr(a' '+A't) y_tdt 1 j =i
(8.12)
m
1
2ni
r 1
m o 1 ~TTr(a -a s-A t)"TTr(a'+a,s) J=i J=I
x ds.
r r n p +b S+B U j=i
Denoting the inside integral in (8.12) by D , we see that its integrand s = -(a'+p)/a>, J J j = l,...,m , p = 1,2,..., and the right set s = (a.-A.t+p)/a., t € L .
function
has
two
sets
of
poles:
i
the
left
set
J
62
J
j
*-
Moreover, in accordance with (8.6), we get that the left poles are simple. Hence from the ( residue \ theory (-l) p of the gamma-function for s = ~(a'+p)/a' we get res T(a'+a's) = , p = 1,2,... . j
j
v
J
)
J
a>
p!
j
Now applying Slater's theorem to the inside integral in (8.12) we obtain that D is equal to the sum of residues of all the left poles, i.e., Ill
m i r—,
> ,
«. /<x a /a
,
r—.
r
U.
CX
\
^
a'
a'
'
r r r « + -± a -A t+ -i P j=i
V
j=l m l—1>[
T^rV j=i j*k
Substituting D
v
»
, k
, j
,
T V "7 a
a
J
a'
J
k
(8.13) ]
p
{-IV
TT~
;
a' k
x
p/a'
k
^
(8.13) in (8.12) and again applying Slater's theorem
we obtain representation (8.10). Thus Theorem 8.1 is proved.
■
Example 8.1. Let us
H[x,y; (<x,l); (0,1); L ,L ] S' t
l
, ..2 (2TTI)
r(a+s)TO+t)x s y \isdt. L.t L s
Then the convergence region of the last integral is described as follows |arg(x)| < TT/2
and
|arg(y)| < TT/2.
In this case according to (8.10) we get
(3 -x -y H[x,y;(a,l);(/3,l);Ls,Lt] = xV^T-x.-y; - ;-l = x°V e e J . 63
Remark 8.1. The above repeated method, which is used to establish representation (8.10), in general case transform the general H-function of two variables (2.2) in terms of the sum of several incomplete double series of the next type
xx vVyVL^ Ss» (xV) (xV)p (x'V (x^y ))q,,
(8.14) (8.14)
p
P
q
q
(p,q)€A
where A is some subset of the set IN2 = IN x IN , where IN denotes the set of
non-negative
integers.
This
situation
appears,
oor example, in
o f u d n o n - the next H-funers.n H[x,y;(a,1,1),(0,1,0),(0,0,1); (8.15) (8.15)
L ,L.] S L s L
♦ + ico i o o +ioo +i»
= — 121 = — {2niV2 {27ii)
Remark 8 . 2 .
JJ J J -ioo -ioo -ioo -ioo
t s r(a+s+t)r(s)r(t)x"y dsdt, r(a+s+t)r(s)r(t)x" y' t dsdt,
Theorem 8 . 1 shows t h a t
(2.3)
can be r e p r e s e n t e d
will
coincide
with
i n t h e form
N2. However,
Re(a) > >0 0.. Re(a)
if the integrand function (8.1),
we note
e(s,t)
then t h e subset A i n (8.14)
that
condition
(8.1)
is
sufficient and in certain other cases the corresponding H-functions can also be represented by the sum of several complete double series.
Example 8.2. Let H[x,y;(a,-l,l),(/3,-1,-1), (0,1,0), (0,0, l);(y,-l,l);Ls,L l);(y,-l,l);Ls,Lt]
= -^— 2
f
f T a-s+t, /3-s-t,s,tl r x-y-'dsdt.
(27ri) J J L *~s+t L
J
L
t s
64
Then above integral converges if |arg(x)| < 7i, |arg(y)| < n
I argix)-arg(y)
and
I < n.
In this case we have a+t-s, 0-t-s,s
1
rm/Sit gij 2ni
1
2ni
r(t)y
x ds y+t-s
W*
a+t+p, /3-t+p dt y+t+p
p= 0
I
(-1)PP|
a+p+t,0+p-t,t 1
y_tdt
I 2ni
p!
y+p+t
p= 0
The last integral for each p = 0,1,2,... is evaluated by two sets of the left poles t = -q and t = -a-p-q. Hence we obtain finally, that our integral is equal to
a+p-q,/3+p+q p=0
E3
)
y+p-q
q=0
a+p+q
q =0
r a
[ ;1v^™)^f+!:r]/
a+0+2p+q, -a-p-q
xr y-a-q
x H (a+/3,1+a-y, l+a;xy,-y), where |x| < 1, |y| < 1 and |xy| < 1/4, |y| < 1/2+1/2/1-4|xy| denote the following functions from Horn's list
(A.Erdelyi
V. 1) i
(8.16)
G (a,0,0';x,y) = ) 1
/
<
m
(a)
m+n
(0)
n-m
65
(0')
n
V^, ■
m-n m ! n !
; G
and H
et al. 1953,
00
(8.17)
H3(a)P>r;x,y)
-
E-
(a)
(/3) 2m+n
'—' m, n = 0
Example 8 . 3 .
m n n ^ X y .
m+n
Let
H[x,y;(a,a,A)
; ( 0 , b , B ) ;L ,L ] 1
(2rri)2J -100
1 0
0
J L
J
-100
Then this integral converges if maxj |Aarg(x)-aarg(y)|,|Barg(x)-barg(y)| V < -^- IaB-bAI.
Hence after the replacement of variables s
= as+At, t
= bs+Bt and
from Example 8.1 we obtain the following representation H[x,y; (a,a,A) ; (/3,b,B) ;L ,L J =
f
B/A x exp -x
-b/A y
x
y
-A/A a/A -x y
■ ) ■
where A = aB-bA * 0, Reioc) > 0, ReifB)
> 0.
Example 8.4. Let us
H[x,y;(a,-1,0),0,0,-1),(0,1,0),(0,0,1); (y,-l,-l),(A,l,l);L
,L 1/2
(8.18) l/2+loo l / 2 + loo
a-s,/3-t, s, t (27li) 1/2-ico 1/2-ioo
Jte(a),
y - s - t , X+s+t
Re((B)>l/2. 66
x y dsdt,
] 1/2
According to estimation (4.10) here we have
a-s,/3-t, s, t e(s,t) = r
exp
[-n(|s|+ |t|-|s+t|j]
,Re(a)-l/2
y-s-t, A+s+t Re(/3)-l/2 X
Consequently
s+t
t
from
Theorem 4.1
-Re(y)-Re(A)+l/2
we
obtain
that
the
integral
in
(8.18)
converges if and only if x,y € IR and
(8.19)
Re(a-y-A)+l < 0,
Now we present
Re(/3-y-A)+l < 0, fie(a+/3-y-A)+3/2 < 0.
H-function
(8.18) in terms of double
series. We
2
consider this question in the four domains of IR : {0<x, {0
1) The
case
0 < x < l ,
and
{Kx, Ky>.
0 < y < 1. Since Re (a), tfe(/3) > 1/2
then
integral in (8.18) can be evaluated twice from the sum of the residues of the left poles of the integrand function, i.e., at first for s = -p and then for t = -q. Finally, denoting the H-function
(8.18) by H[x,y], we
get 00
a+p,/3+q
(-l)p+q
V"
x p yq
y+p+q,A-p-q
p,q=0
a,0 F (1-A,a,0;y;x,y), where
F (a,b,b';c l
x y)=
- I
(a) p+q
(b) (b') p q
(c)
p,q = 0
P q x y
p!q!
p+q
is the first Appell series. 2). The case
0 < x < 1, l < y . Then by the replacement of t by -t in
(8.18), we get 67
-1/2+lco
l / 2 + loo a-s,/3+t, s, -t
H[x,y] = (27Ti)2
y-s+t, A+s-t
-l/2-loo
x"s(i)_tdsdt. y
l/2-loo
Hence the last double integral can be evaluated by residues of the left poles s = -p, t = -£-q as follows
ir
i
ct+p,/3+q
t-l>P*q
V
P -/3-q
x y
r
y+p-/3-q,A-p+/3+q p,q=0
a,0 y"/3Gi(a,3,l+3-y,l-^-A;xfi)l
=r y-|3,A+0 where G
l
is Horn's series (8.16).
3) The case
1 < x, 0 < y < 1. Similarly to the second case we obtain a,0
H[x,y] = r
x y-a,A+a
4) The case
a
G (a,/3,l+a-r,l-a-A;i,y). l x
1 < x, 1 < y. Then the integral in (8.18) can be evalu
ated by residues of the right poles s = a+p, t = 0+q as follows
H[x,y]
a+p,0+q
V ("1)P*
x
-a- P -/3-q
y
y-a-0-p-q, A+a+0+p+q
p,q=0
a,/3 x %
=r A+a+0, l+a+0~r
^F (l+a+0-y,a,0;A+a+0;i,i). l x y
The values of H[x,y] for x = 1, y = 1 are defined by the continuity of the function H[x,y] and they can be obtained from Gaussian theorem (see L.J.Slater (1966)) c, c-a-b'
(8.20)
F (a,b;c;l) = V 2 1
fle(a+b-c) < 0. c-a,c-b
68
summation
§9. Characteristic of the general H-function of two variables
In this section we introduce important notion for classification of the general H-function
of
two variables. Remember
a u+A v = 0 , j
determine
in the
that any equation
j = 1,2,...,m+n,
j
(u,v)-plane
the corresponding
line L
which passes
through the origin of co-ordinates. Since any two proportionate pairs (a ,A ) 1
and
(a# ,A ) = (Aa. , AA. ) J
l
2
J
2
J
i
define
only
one
line
in
the
i
(u,v)-plane, then the exact number of these lines may be less than m+n. We shall call this number by the term of characteristic of the general H-function
of
two variables.
In accordance with
Definition
4.1
of
neighboring lines the projection to axis (Ov) of the point of intersection of line L, and upper half circle with unit radius equals to a . Hence we give the
exact definition of this notation.
Definition 9.1. If among a
in (4.14), j = 1,2,...,m+n, there exist
only r various values, then number r is called the characteristic of the general H-function (2.2) of two variables. If the characteristic of the H-function (2.2) is equal to r, then after combining all proportionate pairs for the function G(s,t) (2.3) we can always write (9.1)
G(s,t) =
r T1 T G (d s + D t), ' k=l
k
k
k
where d D -D d * 0 for k * %J j , and G (T) i s a function of one variable T of k j
k j
k
*
the type
TTr(a '_'
(9.2)
G (T) k
= - ^ n
jk
+ a T) jk
k
T T' r o jk + b jkT) 1
j=l
69
Let
r = 1,
accordance
i.e.
with
a A -A a j k
=0
j k
Corollary
for
4.4
all
the
j , k = 1,2, . . . ,m+n.
corresponding
Then
H-integral
in
(2.2)
diverges. Let now r = 2 . Then t h e H - f u n c t i o n two F o x ' s H - f u n c t i o n
(2.2) breaks into the product
i n t h e a l t e r n a t i v e form ( 2 . 1 ) a s
H|x,y; (a,a,0) L
m
,(a',0,A)
1
2
;
m
(/3,b,0)
n
of
follows
,(/3',0,B) l
n
2
; L , L.l s tJ
(9.3) = H[x;(a,a)
;(/3,b) ;L] n s
m 1
H[y; J
H f x , y ; (a, k a , qn a )
;(/3\B) mn
m
2
, ( a ,pA,hA) ^
1
(9.4)
(a',A)
1
;L.]. t 2
;
m 2
(/3,kb,qb)n , O ' ,pB,hB)n ; L Q , L J n R i 2 ° -I
= J_ lAi
H
[xh/A y-p/A; L
(«,a)
;0.b) m
; L kC+qC
i
_1 J
x H | x - q / \ k / A ; (a'.A) AP'.B) ; L +. _ 1 , L m n pc+hC J where k , h , p , q G R, A = k h - p q ^ 0 and ( a , k a , qn a ) means (a , ka , nqa ) , . . . m
(a
m
, ka
l
,qa ) , and so on. The e q u a l i t y m ^ m l
(9.4) with the l i n e a r property
Let r = 3 . Then from in the next
1
but
1
(9.2)
is
(9.1)-(9.2)
(6.4).
we can a l w a y s w r i t e t h e
form
H[x,y;
(9.5)
1
l
is evident,
l
t h e c o m b i n a t i o n of
=H
(9.3)
[w
(a,a,A)
; (0,b,B) ; L , L,. ] m n s t
(^,c,e)
m 1
(5,E,E)
n
, (a',a',0)
.(a'^O.A*)
2
;
m 3
,(/r,b',0) 1
m
2
n
70
, (/3",0,B')
n
; L' 3
,L! 1 s t J
H-function
m
m
1
m
2
3
7 7 T[y + c (s+t)] "f7 r(a'+ a's) 7 7 r(a" + A't) r
J
j=i
(27Ti)2
J
J
j=i
n
J
1
J
j=i
n
J
n 2
3
L;L' 7 7 r[5 + E (s+t)] 7 7 ro'+ b's) 7 7 r(£" + B't) x x 7y dsdt, 1
i
where x = x y , y = x y ; e, E, a* , A' , b' , B* € R; y, 6, a' , a' * , |3' , £'' € C and they are defined
from a, a, A, £, b, B ; L' , L' are new
contours. We shall study this function in detail in Chapter II. Further the G-function of two variables defined by O.I.Marichev and Vu Kim Tuan (1983) can be an example of the H-function (2.2) of two variables with the fourth characteristic. This function is represented as
follow (IK
f(
m ,n ;m ,n ;m ,n ;m ,n |(a ^ 1 J
1
2
2
3
3
4
4
pi,qi;p2,q2;p3,q3;p4.q4
(b
(1)
)
, (3). , (4).
, (2)x
),(a (b
), (a (2)
)
(b
(3
), (a
) <4
(b >)
')
x,y
(9.6) y +loo y +ioo 1 2
1 (27ii)
$ (s+t)$ (s-t)$ (s)$ (t)x "y _tdsdt, 2
1
V -ico
2
3
4
r2-- 1 0 0
where b
(9.7)
* k
(T)
1
+T, .
= r b
n +i k
+T,
v< k
b m
^
(k)
k
,a ( k ) + x, P
-
+x, 1-al -x,
* <
1-b
k
k
>
m +i k
-T,
-
k
, 1-a n
-T
k
,l-b ( k , -T
1,2,3,4. Finally, the H-function (1.14), defined by P.K.Mittal and K.C.Gupta (1972) is an example of H-function (2.2) with a characteristic r £ 4.
71
CHAPTER II. THE H-FUNCTION OF TWO VARIABLES WITH THE THIRD CHARACTERISTIC
§10. Definition and Notation In Chapter I, §9 we have
introduced the notion of the character
istic of the general H-function of two variables and showed that this function is of type (9.5) when its characteristic is equal to three. In this chapter we use the following notation of the H-function with the third characteristic.
Definition 10.1. H-function H(x,y) of two variables with the third characteristic is called the next function (1)
(
m ,n :m ,n ;m ,n 1 1 2 2 3 3 x,y P 1 ,q i :p 2 ,q 2 ;p 3 ,q 3
(a
UK
,a
(2)
f
): (a
(2K
,a
,
(3K>
(3)
); (a
,a
)]
(/3(1),b(1)):(/3(2),b(2));0(3),b(3)) q
q
i
i
q
2
q
2
q
3
q
3
(10.1)
$ (s+t)$ (s)$ (t)x ^ ^ d s d t , 1
{2niV
2
3
L L 2
1
where
(10.2)
$ (s+t) =
m n rtr(p < 1 ) + b < 1 ) (s + t)]TTr(i-a < 1 , -a ( 1 , (s + t)| J j j=l Ij Jj = l I J '
Y\ j=n +i
rfa!1,+a;i,(s+t)] TTrfi-P^-b^ts.t)) ^
J
j
72
' j=m +i l>
'
m
n
urfrw?.) rp-KS"') 2 /
(10.3)
$ (s) 2
\
p
2
q
-TTrf«(2>+a(2)s) 7 7 rfi-p (2, -b (2, sl
m
n
rpK*."'] 17H"-i"«) 3
(10.4)
$ (t) 3
/
3
N
P,
q3
rrr( a ; 3, + a; 3 't)rTr(a- P ; 3> -b; 3 't)
j=n + i v
J
J
J
> j=m + i ^
J
'
where x and y are not equal zero and an empty product is interpreted as unity. Also, the non-negative integers m that O ^ m
k
^ p
Greek letters a
(j = 1,2
k
(k=l,2,3) are such (k)
,b
are all complex numbers.
is in the s-plane and runs from f -ioo to f +ioo and poles
of
(j = l,2,...,n2)
T 1-a
from
the
-a
(s+t)
poles
(j = l,2,...,n ),
rf/3(1)+b(1) (s+t)l
of
m ), riy 2) +b| 2) s] (j = 1,2,...,m ), when t € L . *
^ J
The contour L
J
J
2
2
is in the t-plane and runs from f -ioo to f +ioo and 2
separates
are all Fpositive,
(k )
, £
the
r(l-a<2)-a'2)s]
,q
^ q ; Latin letters a
k
The contour L separates
,p (k)
,0 ^ n
k (k )
,n
the
rflV 3 ) -a^ 3 ) tl
2
poles
r(l-a (1 *-a (1} (s+t)1
of
(j = l,2,...,n3)
(j = l,2,...,m ), r(/3(3)+b(3)tl *
Y,
■*
■*
from
the
(j = 1,2,
poles
2
(j
=
l,2,...,n),
of
T^ ( 1 ) +b ( 1 ) (s+t)l
m ) when s € L .
J
Remark 10.1. If we take m
=n
=p
=q
=0
in (10.1) then
H(x,y) breaks into the product of two Fox's H-functions (1.5)
73
, 2
2
3
2 *3
3
, 2
p
2 n
*2
(/3(2>,b<2,); (P
,a
(0
2
2 (2
p
\b
3 n
*3
p
(0
3
3 ,3)
(3)
3
P >q
)
)b
(a
m ,n
)
p
3 (3)
,
)
2 (2)
,a
P 2
(2) ( 2 ) .
(a
, (3) ( 3 ) .
); (a
p 2
3
m ,n
(2).
,a
p
0,0 :p ,q ;p n,q x,y n *2
(2)
(a
0,0 :m ,n ;m ,n
)
<3K
,a p
3 (3)
,b
)
3 (3)
)
Also, if in the integral (10.1) were place s+t by s and t by t then we obtain the following formula in the case
m = n = p = q = 0
&
2
f 1 1
3
3
i
**3
(|3
3
(1)
l
r (a
l
P.
P >q *i
,b q
P
(P
):
3
(1),
,
b
3
(3)
(a
3
P >q n
*3
( 3 ) q
i
m ,n
P1
P
3
(3)
,
n ,a )|
2
i (1)
i
M
*2
(a(3), a<3)))
(1
C0(1).b
l
p
i (1)
q
m ,n
(IK
,a ):
p
x,y
p ,q :0,0;p ,M q M *i
(1)
(a
m ,n :0,0;m , n
2
(P
3
>
(3),.,
,a P
3 ,3)
)
3
P
) 3
,b(3))
Similarly, we have ,
i
*2
, (2)
): (a
l
(1) ,(1)
2
): (P
<2>
= H P >q
(/3U,,b
2
(1),
,b
f
(2),
(2)
(a
m ,n
1
(2K
,a );
l
(1) r(a(1) ,a n )|
m ,n l
O
2
(IK
,a l
1 1 2 2 p ,M q :p M ,q ;0,0x,y *i
(1)
(a
m , n :m ,n ;0,0
(2) . A
,a
)1 2
2
P »CI
(|3(2,,b
(2)
*2 ^2
Remark 10.2. In the case, when m = 0 (or n = 0 ) , the H-function l
(10.1) coincides with R.S.Pathak S.L.Kalla
the functions
(1970), K.K.Chaturvedi (1971),
S.L.Bora
l
of two variables
and A.N.Goyal
and S.L.Kalla
(1970),
R.U.Verma (1971), M.Shah (1973a,b) (see also (1.11)). 74
introduced by
(1972), P.C.Munot and R.K.Saxena
(1971),
§11. Convergence Theorems
Now f o r d = f + f 1
(11.1)
, d = f , d = f
1
2
K = -^-
2
1
3
2
k k ^ , (k)^ „
(k)
m
n
>b
+ Fa
J
and k = 1 , 2 , 3 d e n o t i n g &
p „
-
q
k
y
a
J
j=i
(k)
„
-
_
/i = Re
j=i
^k
(k)
j=i
„
0
j
j=n +i k
(k)
, (k)
b
J j=m +i k
P (11.2)
k
Y
+ d
j=i
9 k
„
(k) j
U
J=l
p - q
k ~ ^(k) j J=l
*k
^k
and making use of the general asymptotic estimation (4.10) obtained in §4 we have
|* (s+t)$ (s)$ (t)x sy"t
(11.3.)
« S exp
|U+V|+K
[- ■(-.
"M2
"**1
x |u+v+i |
|U|+K |v|| + uarg(x)+varg(y)
|u+i|
^3
|v +i| 2
2
where H = const, s e L , t e L , u = Zm(s), v = J/n(t), u + v 1
2
from Theorem 3.2 and Remark 3.3
Theorem 11.1. The integral
—>oo. Hence
we obtain the following results.
in
(10.1) converges
if,
f |arg(x) | < 7r(/c + K ) , 1 1 2
(11.4)
Urg(y) | < 7T(K + K ) , |arg(x)-arg(y)| < TT(K + K ) .
7f we replace one
(>) for at least
the symbol one above
(<) in conditions inequality,
diverges.
75
then
(11.4) by the the
integral
opposite
(10.1)
will
Consequence
11.1
If
K + K > 0,
n
integral
1
(10.1) converges
for
all
K + K > 0,
2
1
real
K + K > 0, 2
3
positive
variables
then
the
3
x and y.
Further, here we establish the following auxiliary lemmas for the wording of more strong criterion of the convergence of the
integral
(10.1) in the case x,y e (R+ .
Lema 11.1. Let
B,C,a,b,c 6 R and i = "/-T . Then
lu+v+i | a |u+i | |v+i|° exp(Bu+Cv)dudv < oo
(11.5)
if
all
and only
if
f 2sgn(B) + sgn(a+b+l) < 0 , (11.6)
2sgn(C) + sgn(a+c+l) < 0 , 2sgn(B) + 2sgn(C) + sgn(a+b+c+2) < 0
Proof. It is showed by Lemma 4.1, Chapter I, §4, that the integral (11.5) converges
if and only
if one of
the following
four
conditions is true:
1)
B < 0, C < 0.
2)
B < 0, C = 0, a+c+1 < 0.
3)
B = 0, C < 0, a+b+1 < 0.
4)
B = C = 0, a+b+1 < 0, a+c+1 < 0, a+b+c+2 < 0.
Evidently it is equivalent to the inequalities (11.6).
Lemma 11.2. Let all (11.7)
A,B,C,a,b,c € R , i = V^l and
Q(u,v) = |u+v+i|a|u+i|b|v+i|° exp(A|u+v|+B|u|+C|v|).
Then
76
sets of
(11.8)
fi(u,v) dudv < co
<=»
ur f 2sgn(A+B) + sgn(a+b+l) < 0 , 2sgn(B+C) + sgn(b+c+l) < 0 , 2sgn(C+A) + sgn(c+a+l) < 0 , 2max jsgn(A+B)+sgn(B+C),sgn(B+C)+sgn(C+A),sgn(C+A)+sgn(A+B)t + + sgn(a+b+c+2) < 0. Proof. As is known from Chapter I, §4, three
lines represented by
the equations u+v = 0, u = 0, v = 0, separate the (u,v)-plane into six nonintersecting sectors W ,...,W . In the sector °
1
6
W = { ( u , v ) € IR , u > 0, v > 0} from Lemma 1 1 . 1 we h a v e Q ( u , v ) dudv
lu+v+i | a |u+i | |v+i|c exp(A|u+v|+B|u|+C|v| ) dudv < co o o if and only if ( 2sgn(A+B) + sgn(a+b+l) < 0 , I 2sgn(A+C) + sgn(a+c+l) < 0 ,
(11.9)
[ 2sgn(A+B) + 2sgn(A+C) + sgn(a+b+c+2) < 0 Further, as in §4, if we replace u+v by v sector W
and -u by u
then
= {u,v € IR , u < 0, v > 0, u+v > 0} breaks into W . Hence 2
Q ( u , v ) dudv
1
Q(u +v , - v ) du dv i
l
l
1 1
77
the
| u +v + i | b | u + i | a | v - i | c e x p [ ( A + B ) |u | + (B+C)|v 1
1
1
1
1
I )du dv i
l
l
< oo
O 0
if and only if
f 2sgn(A+B) + sgn(a+b+l) < 0 , 2sgn(B+C) + sgn(b+c+l) < 0 ,
(11.10)
2sgn(A+B) + 2sgn(B+C) + sgn(a+b+c+2) < 0
Similarly,
fi(u,v) dudv < oo
(11. 11) W
2sgn(C+A) + sgn(c+a+l) < 0 , 2sgn(B+C) + sgn(b+c+l) < 0 , [ 2sgn(C+A) + 2sgn(B+C) + sgn(a+b+c+2) < 0 Note that the double integrals from the Q(u,v) on the domains W , 4
W , W 5
converge provided conditions (11.9), 6
respectively.
Finally,
(11.10),
(11.11) hold true
the set of these conditions
is equivalent
conditions (11.8). This completes the proof of Lemma 11.2. Lemma 11.3. Let, as in Lemma 11.1, all
(11.12)
sup
i lu+v+i |a|u+i |b|v+i |cexp(Bu+Cv)i < oo
J
f 2sgn(B) + sgn(a+b) ^ 0 , 2sgn(C) + sgn(a+c) ^ 0 , 2sgn(B) + 2sgn(C) + sgn(a+b+c) < 0 Proof. At first, note that the function
(11.14)
Q (u,v) = |u+v+i| |u+i| |v+i| exp(Bu+Cv)
78
■
B,C,a,b,c € !R and i = -/-F. Then
u,v > 0 v
(11.13)
to
is continuous in the plane R . Hence the inequality (11.12) holds valid if and only if
(11.15)
lim Q (u,v) < oo o
2
for u +v
2
—> oo .
Further, it is not difficult to obtain the following statements:
i) Let B = 0, C < 0. Then the inequality (11.15) is true if and only if
a+b < 0.
ii) Let B = C = 0. Then lim Q (u,v) < oo o
2
for u £ v, u +v
2
—> oo ,
if and only if a+b ^ 0 and a+b+c ^ 0.
Now from the above statements i) and ii) we can prove the inequality (11.12) holds valid if and only if one of the following four sets of conditions is true 1)
B < 0, C < 0.
2)
B < 0, C = 0, a+c ^ 0.
3)
B = 0, C < 0, a+b ^ 0.
4)
B = C = 0, a+b < 0, a+c < 0, a+b+c ^ 0.
But it is equivalent to the conditions (11.13).
Lemma 11.4. Let
(11.16)
Q(u,v) be defined
sup
by (11.7).
Q(u,v) < +oo
Then
«=»
U, V > 0
2sgn(A+B) + sgn(a+b) ^ 0 , (11.17)
2sgn(B+C) + sgn(b+c) ^ 0 , 2sgn(C+A) + sgn(c+a) ^ 0 , sgn(A+B) + sgn(B+C) + sgn(C+A) + 2sgn(a+b+c) ^ 0.
79
This Lemma can be proved by analogy with the proof of Lemma 11.2. Here it is sufficient to note that if A+B ^ 0, B+C ^ 0, C+A ^ 0, then the condition
(11.18)
2 maxisgn(A+B) + sgn(B+C), sgn(B+C) + sgn(C+A),
sgn(C+A) + sgn(A+B)l + sgn(a+b+c) ^ 0
is equivalent to
sgn(A+B) + sgn(B+C) + sgn(C+A) + 2sgn(a+b+c) ± 0.
Now from the asymptotic estimation (11.3) and Lemma 11.2 we obtain the following theorem about the convergence of the integral in (10.1).
Theorem 11.2. The integral variables
x and y if and only [
(11.19)
in
(10.1) converges
for
real
positive
if
2sgn(K + K ) + sgn(ji + jn.-l) > 0, for k,j = 1,2,3 and k * j;
2min «|sgn(K + K )+sgn(K + K ), sgn(/c + K )+sgn(»c + K ), ^ 1 2 2 3 2 3 3 1 sgn(/c + K ) + sgn(K + K ) \ + sgn(a + u. + a - 2) > 0. 3
1
1
2 J
1
2
3
We conclude by remarking that Theorems 11.1 and 11.2 immediately follow from Theorems 3.1 and 4.1 respectively, as particular cases.
§12. Reduction formulas for the H-function with the third characteristic
The H-function with the third characteristic (10.1) as a particular case, possesses all properties in § 5 — § 8 for
the general H-function
(2.2). In this section we consider the cases when the H-function
80
(10.1)
of two variables reduces to Fox's H-function These
properties
are usual
only
(1..5 of one variable.
for the H-function
with
the third
characteristic. Here we assume additionally integral
that
in (10.1) are the vertical
the contours L lines with
and L
i
in the
2
real parts c and C
resperalvely, i.e.
L
l
s LL
, c = {|1sseeC C,
Re(s=cc
1, J1,
L L2 =L Lcc = =j jt t€ € C, C, Rei)) Rei)) = =C C 1. 1. 2 =
First we establish auxiliary result.
Lemma 12.1. Let c > 0, C > 0. Then
(12.1) (12.1)
(27Ti)
(12 2) (12.2)
J J L C Lc
1
1
formulas
are
valid
0(x)r(T)(x+y)~TTdr, >(s+t)r(s)T(t)x"V t dsdt = -!— 0(x)r(T)(x+yr
— 1 — —1— 2
the following
2711 J L C+c
[*Cs+t)x]__[f 00((TT)) tmax(xty)]^ tmax(xty)]^ ff [*te+Uxy-tdsdt == __J y -tdsdt tt TT
(27ii )' JJ (2TTI)' L
27ri JJ 27ri
Js s J
L
C
c
"C+c
where T(s) is
the gamma-fund gamma-fundtion, ion,
the ineegrals
are convergent
{in
0(T) is some function (12.2) it
such
is assumed that
that
all
of
x,y > 0).
The proofs follow from the change of variables s by s , s+t by t , after which the following integrals are used r(p-s)T(s)z"sds = r(p)(l+z)fpP,
—— ? —— 2711
L
1
c ('
s f _ J I z"z" s !S
p ~\ ,
—T T " TFiTids (p-s)s ds " I 2711 2711
0 < < c c < < ReRpp; ReRpp;
J
II
c c
if z < 1,
P [[ P"V .. P _1 zP p ,, if if z z> > . . 81
0 < c < e(Reip). 0< c<* p).
The last two equalities are easily obtained with the help of the residue theory (see, for example, O.I.Marichev
(1983)).
From (12.1) it follows immediately the following reduction formula of the H-function (10.1) to the Fox's H-function (1.5) of one variable (a ,a ):
m,n:1,0;1,0
p
p,q:0,l;0,l x,y
P
(0 ,b ): (0,1);(0,1) q
q
(12.3) m+1 ,n
= H!p,q+i
(a ,a ) p P
x+y
(0,1), (0 ,b )J q
Here for
q
y
simpleness we replace m ,n ,p ,q
by m,n,p,q,
respectively.
Similarly, as a consequence of the equality s = r(s+l)/r(s) it follows from (12.2) that (a ,a ): (1,1);(1,1))
m,n:1,0;1,0
P
p,q:l,l;l,l x,y
P
(0 ,b ): (0,l);(0,i: q
q
(12.4) (a ,a ),(1,1) m+l,n p P max(x,y) p+l,q+l (0,1), (0 ,b )J q
(x,y > 0 ) .
q '
Naturally we will call (12.3), (12.4) respectively the sum and maximum properties for the H-function (10.1) with the third characteristic. Now we consider reduction cases for this H-function when y = x. Lemma 12.2. Let
equality
(12.5)
is
f(x)g(x) = h(x) for
all
x > 0. Then
true 1
^(s+t)f*(s)f*(t)x"s_tdsdt
(27ii r L
CLc {r)h (T)X dx,
2ni "C+c 82
the
following
where f ,g ,h
are
respectively
the Mel 1 in
transforms
of
the
functions
that
all
integrals
f,g,h
(12.6)
f (s)
Moreover,
the
functions
are
f(x)x
assumed
dx.
to
be such
converge. The
proof
of
(12.5)
is obtained
variables T = s+t along with
by again
the complex
using
the change of
convolution
for the Mellin
transforms 1 2ni
h (z)
Example (1+x)
12.1. If, in Lemma
, then h(x) = (1+x)
f (z-t)g (t)at.
12.2, we set f(x) = (1+x) , g(x) =
and consequently
Rein)
f*(s) = r(fi-s)r(s)/r(jz), with
* * g and h similarly
expressed.
Hence
> o
we obtain
the reduction
formula
p
p,q:l,l;l,l
O
p
'
,b ): (0,1);(0,1) q
T{u)r{v) r(u+v)
tt-vtD)
(a ,a ): (l-ji.l);
m,n:1,1;1, 1
q
m+l,n+l "* p+l,q+l
w
(a ,a ) , { l - u - v , l ) } p
p
'
(o.i), o q
Example h(x) = (l-x)^
12.2. Let +y 2
Reiii),
g(x) = (1-x)^
, where (l-x) a , if x < 1, (1-xr = , if x > 1.
Then, since
f (s) = r(fi)r(s)/r(M+s), 83
> 0.
q
f(x) = ( l - x ) M 1 ,
0
Re{v)
,b )
Rein)
> o
1
and
then
from formula (12.5) we have (a ,a ): (jn. 1); iv, 1))
m,n:1,0;1,0
p
H p,q:l,l;l,l
p
'
(0 ,b ): (0,1);(0,1) q
q
(a ,a ), (iLt+i^-1,1 )^
r(jLL+i^-l) j, m+l,n r(ji)r(v) H p+i,q+i
p
p
(0,1), o ,b ) q
Example
Re(fi), Re(v),
'
Re{n+v-l)
> 0.
q
12.3. For generalized hypergeometrie functions
(1.2) the
following formula is well-known (see A.Erdelyi et al. (1953)) F(M;x) F ( V 5 X ) =
(12.7)
F (M2,^-l);
4xl
where A(k,a) abbreviates the array of k parameters a a+1 a+k-1 k' k ' ' " ' * k Further for convenience we use the notation (a)
T(a )...T(a )
(0)
r(s )...ro )
1
0,...,0 1
i
q
Then for the hypergeometric function
p__
F
q
(a) ; (b)
;-z
we have the
P p+i | _ p p+i J following representation (see O.I.Marichev (1983), A.P.Prudnikov et al
(1989))
(12.8)
F P
[(a) ;(b) ;-zl p+i |_ p p+i j
r f ba ,...,b p+1
[ aa L
a i
(12.9)
x
s-i „ F P P+1
1
1
I
f a -s,..., aa -s,s -s,s ] -Sj p |z ds, -s, . . . , b b- -s s J
J -251 J1 *b J
L
p+i
P
[r\ t^\ 1j _|" b , . . . ,b ,a -s, . . . ,a -s,sl 1(a) ;(b) ;-x dx = T l' p + i' l P a , . ,,a ,b -s,...,b -s l P P +1 i_P + I p i_i L J L a a.b-s b -s J 84
Consequently,
from
(12.5),
(12.7),
(a ,a ): p P
m,n:1,0;1,0 (12.10)
H
p,q:0,2;0,2
(12.9)
get
—
;
—
(|3
q
,b
q
):
(0,1-/LI);(0,1-I>)
(-JLI-I>,2), (a ,a )
m+l,n+l ■• p + l , q + 4
p
p
(0,1), (0 ,b ),(l-fi,l), (1-1^,1), (-II-I>,1) q
q
is shown above, any equality of type
corresponding (10.1) with
reduction formula of the
third
hypergeometric series
F
(12.7) shall determine
(12.10) for
the H-function
particular, for Gaussian
we have the following relations
V-A 1
type
characteristic. In
' A,n;
F
(12.11)
)
x,x
u
As
we
X
V-A,i>-/i;
2 1
0
F
X
2 l
X
v
;
>
r
A, H (12.12)
F
X
2 1 [A+ii+1/2
2 1
A+/I+1 / 2
A.fl
2 1
2 1 A+ii-
A+tx+-^;
2 1 A+LI- — ^ 2
; *
J '
2 1 A+fx-
3
2
( 2A,2ii-l,A+fi-l 3 2[A+ii-—
'
2 1
- Xf *
; J
—-A-M
1
F 3 2
^
2
^
2
2
A+LI+ — , — - A - i i ^ 2 2 ^
85
;
I
J
' X
2 A+J1+— , 2 A + 2 i i - l ;
;
(12.15) =
;
F
.
— -A — 2 ' 2
F
2 1 A+ii+ -
2A,2ii,A+fi =
;
J
;
F
A,M ;
f F
X 2 A4 i i + l / 2 , 2 A + 2 / j i;
' X
F
X
F
3
;
A,ii-1 (12.14)
F
;
F
F
=
X
•
A,M ; (12.13)
2A,2ii,A+ii
i ]
,2A+2JLI-2;
}
[A-— ,fi-— ;
1
2
F
F 2 1
^
2
X
1
2 1
A+jLi- -
'
^ 2
(12.16) f
2A,2/i,A+n
;
F 3 2
A+u+— ,2A+2fx-l;
(1 A + — ,ii- — 2 ^ 2
F 2 1
J
(A+ — , u + — ;
;
2
F 2 1
A+/1+
^
2
X+V+—
; J
(12.17) f 2A+1,2/I,A+M
;
F 3 2
(A.A+
(12.18)
2 1
A+n+ — ,2A+2fi;
— 2
F,
} X
• X
2 1
2A+1
>
V.M+-JF
(A+/1, A+/1+ — ; ] x 1 [ 2A+2fi+— ;
= F
X
2
2/1+1
Here (12.11) is a well-known formula obtained by twice applying the Boltz's formula
f a,b ; ) z
F
f a,c-b ; = (l-z)' a
z-1
2 1
2 1
and
F
(12.12)-(12.18) are Slater's theorems
(see L.J.Slater
(1966) and
also H.M.Srivastava and P.W.Karlson (1985), formulas 1.3.34-1.3.42).
Now from (12.5)) and (12.11) - (12.18) we obtain
the corresponding
reduction formulas for the H-function (10.1) of two variables
(a ,a ): (l-v+jbi+X,l); (l-v+A,l), (l-v+jbi,l)l
m,n:l,l;l,2 (12.19)
H
P
p,q:l,l;2,2
(0 ,b ) q
A,fi, y-fi-A
=r
H
v-A,
v-\±
p
'
(0,1)
;(0,1),(1-v.l)
q
m+l,n+2 p+2,q+2
(l-i^+A,l), (1-V+JLI,1), (a ,a )] p p '
(0,1), (0 ,b ), (1-1^,1) q
86
q
(a ,a ):
m,n:l,2;l,2 p,q:2,2;2,2
( 1 - A , 1) , ( 1 - j i , 1)
;
( 1 - A , 1) , ( 1 - j i , 1)
(0 ,b ): ( 0 , 1 ) , ( — - A - M , 1 ) ; ( 0 , 1 ) , ( — q
q
2
)
-A-jx,l)
2
A,M,A,M,2A+2/LI
(12.20) 2A , 2 p i , A+/J, A + f i + — j
X
( 1 - 2 A , 1 ) , (1-2/1,1), ( l - A - ^ , 1 ) , (a
m+l,n+3 p+3,q+3
u H
(0,1),(/3
,b q
(a ,a ):
m, n : 1 , 2 ; 1 , 2 p,q:2,2;2,2
,a p
)
1
P
), ( — -A-^1,1), (1-2A-2JLI,1) q
2
( 1 - A , 1 ) , (1-jix, 1 )
;
( 1 - A , 1 ) , (1-JLX , 1 )
)
x,x (/3 , b ) : ( 0 , 1 ) , ( 3 / 2 - A - / L t , l ) ; ( 0 , 1 ) , ( — q
q
2
-A-^,1)
A,fi,A,fi,2A+2/i-l (12.21)
=r
2A,2/j,A+fi,A+^-
X
u H
—
J
(1-2A,1), (1-2/1,1), ( l - A - / i , l ) , (a
m+l,n+3 p+3,q+3
( 0 , 1 ) , (/3 , b a
(a
m,n:1,2;1,2
,a
):
p
p,q:2,2;2,2
,a p
(1-A,
)
)
p
), ( — -A-/1.1), (2-2A-2/i,l) a
1),
2
( 1-JLI, 1)
'
;
(1-A,
1) , (2-JLI, 1)
} '
p
((3 , b ) : ( 0 , 1 ) , ( 3 / 2 - A - f i , l ) ; ( 0 , 1 ) , ( 3 / 2 - A - J L I , 1 ) q
q
A,/I,A,/I-1,2A+2/LI-2
(12.22) 2A,2/i-l,A+/i-l,A+/i--
( 1 - 2 A , 1 ) , (2-2jLt,l), ( 2 - A - f i , l ) , (a
m+1,n+3
p,q:2,2;2,2
(0,1), O
(a ,a
m,n:1,2;1,2
,a p
x H p+3,q+3
P
):
q
,b
q
)
), ( — -A-fx,l), (3-2A-2*i,l) 2
( 1 - A , 1 ) , ( 1-JLI, 1 )
P
) p
; ( — +A, 1 ) , ( — +|Lif 1 ) 2
2
^
x,x (0 , b ) : ( 0 , 1 ) , ( — - A - M , 1 ) ; ( 0 , 1 ) , ( - — +A+/i,l)
87
-A.--M,
A , jLl,
(12.23) A-JLI+ — , f i ^
(_L-A+ii,l)f ( J--M+X,l),( — , l ) , ( a
m+1,n+3 x H p+3,q+3
2
2
2
,a P
)) P
( 0 , 1 ) , ( 0 , b ) , ( — - A - | U , 1 ) , ( - — +A+JU.1) q
q
2
2
(a , a ) : ( ^ - - A , l ) , ( — - f i , l ) ; ( — - A , l ) , ( — - ^ 1 , 1 ) ) p p 2 2 ^ 2 2 '
m,n:l,2;l,2 p,q:2,2;2,2
(0 , b ) : ( 0 , 1 ) , ( — - A - f i , l ) ; ( 0 , l ) , ( — - A - / i , l ) q
q
2
2
A - — , n - — ,A+l/,*i+ — ,2A+2ji-l' (12.24) 2A,2fi,A+ji,A+jLi- —
m+l,n+3 x H p+3,q+3
(1-2A,1), (l-2jn,l), ( 1 - A - M , 1 ) , (a ,a ) ) p p ( 0 , 1 ) , (0 , b ) , ( — -A-/U, 1 ) , ( 2 - 2 A - 2 M , 1 ) J q
q
2
(a , a ) : ( J - - A , l ) , ( — - J I , 1 ) ; ( — - A , l ) , ( — - , 1 , 1 ) ] p p 2 2 2 2 '
m,n:1,2;1,2 p , q : 2 , 2 ; 2 , 2 x,x
(0 , b ) : ( 0 , 1 ) , ( — - A - f i , l ) ; ( 0 , 1 ) , ( — -A-,1,1) q
A+—
,H~—
q
2
2
, A + 1 / , M + — ,2A+2/i"
(12.25) 2A+l,2jn,A+fi,A+^+ ■
m+l,n+3 p+3,q+3
(-2A,l),(l-2/i,l),(l-A-/i,l),(a p
( 0 , 1 ) , (0 , b ) , ( — - A - j y i , l ) , ( l - 2 A - 2 f i , l ) q
m,n:l,2;l,2 p , q : 2 , 2 ; 2 , 2 x,x
,a ) p
q
2
(a , a ) : ( 1 - A , 1 ) , ( — -A, 1 ) ; (l-^i, 1 ) , ( — - j i . l r t p p 2 2 ^ ' (0 , b ) : q q
( 0 , 1 ) , (-271,1)
88
;
( 0 , 1 ) , (-2ji, 1)
A , J L I , A + — ,li+— (12.26)
=r
,2A+2|i+ —
\+li, A+fi+ — 2A+1, 2/n+l
(1-A-/LX, 1), ( —-A-[z,l),(a ,a ))
m+l,n+2 x H p+2,q+2
2
P
P
(0,1), (0 ,b ),( 4--2A-2fi,l) I q
q
2
'
§13. The G-function of two variables and its special cases
In this paragraph we introduce the G-function of two variables, which is a special case of the H-f unction (10.1) of two variables, when all coefficients
a
, b
, k
1,2,3,
are equal to 1
,
(13.1)
1
1
2
2
3
3
s
M
l
*2
M
2 *3
M
):
(0(1)):
3
m ,n :m ,n ;m ,n
H
1 1
2
2
3
3
*1
l
M
*2
2
1
(2),
*V
M
0(2));
(a (|3
(3K 3
^
) }
(3),
(a(1),l):
(a(2),l);
(a(3),l) 1
3
(/3(1),1):
(0(2),1);
(0(3),1)
* (s+t)¥ (s)¥ (t) x sy
(27Ti) 2 , I-
.
);
x,y
p ,q :p ,q ;p ,q M
,
(a
x,y
p ,q :p ,q ;p ,q *1
(1).
(a
m , n : m , n ;m , n
1
2
\isdt,
3
I
2
"l
where
77 (13.2)
* (x) =
ro! k ) + x)
^r(i-ajk)-x
- ^
j= l
Pk
k
k
77 j=n
+i k
r(a (k) +x) J
k =
7 7 rd-/3 ( k ) -T) j=m +i k
89
J
1,2,3.
Here, as in Definition 10.1, for the H-function with the third characteristic, the contour L runs from f -i« -i« to f +ioo, +i«, the contour L runs € • * rf2 +i■ • tl i ' HH t-planes t i tf fa-i' to t f ' in ' the th complexs1 1 * ' respectively. f i 1 i-t ft , from f and Likewise from « , to f2+i~ m the complexsand t-planes Likewise theseV icontours satisfy^ the known condition of respectively. poles separation with these contours satisfy^ the known condition of poles separation with In particular, if i^ = 0 (or i^ = 0) our G-function (13.1) coincides with the Agarwal's G-function (1965) and Sharma's S-function (1965) (see also (1.8)). The
convergence
theorems
for the G-function
(13.1)
immediately
follows from Theorems 11.1 and 11.2. Here in accordance with (11.1), (11.2) we denote (13.3) (13.3)
cc = = m m + + n n -k k k k k k
pk
Pk+q p +qfc —■=— , V^ . 2 2
qk
(k) (k) ((kk)) y y = Re R e V> a ct -- V) 0 0 + (d — ) )(p -q ), ), kk == 1,2,3, 1,2,3, j j k 2— k (p k -q
(13.4)
kk
\ L^ L*
^ j=l j=i
L^
j=l j=i
J'
where d = f +f , d = f and d = f . Consequently, according to (11.3) 1
1
2
2
1
we obtain the estimation
3
2
n
I* (s+t)¥ (s)¥ (t)x"sy't| - S 2 exp \-nlc 1
1
2
3
LI L I1 1 -r
J
* ' 1
+ uarg(x) + varg(y) |u+v+i|
-r1x -y
| \i+i |
J
»
o
*
|u+v|+c |u|+ c Ivl] 22
33
J
-y -y |v+i|
+ uarg(x) + varg(y) |u+v+i| | \i+i | |v+i| for u = Jm(s), v = Izn(t) as Izn(s), J/n(t) —> <x>,s € L , t € L . 1
2
1
2
for u = Im(s), v = Jm(t) as Izn(s), J/n(t) —> oo,s4- V»*a € /"" L f,n nt n +€ i L . Mr\t F ^ FA rr l \ / £ i +•}■»£» r i r \ r r o c n r \ r » r l i n n 4" V» A A T - « m e * -f n r An Mr\t F \ FA
rr l \ / £ i
+•}■»£»
rir\rrocnr\r»rl i n n
Theorem 13.1. The integral
4" V» A A T - « m e *
- P A T - 4- V»*a C
in (13.1) converges
f n n n + i An
if, if,
f |arg(x)| < 7i( c 22)) , , 7i(CCii + c (13.5)
|| |arg(y)| < rct^ 7r( C i + C c33) , (( |arg(x) |arg(x) -- arg(y)| arg(y)| << TT(C TT(C22 ++ cc33)) . . 90
f 11
i \
f 11
i \
If
we replace
one (>) for at least
the symbol
(<) in conditions
one above inequality,
(13.5) by the
then the integral
opposite
(13.5) will
diverges.
Consequence 13.1. If n
integral
(13.1) converges
c +c
> 0, c +c
1 2
1 3
for all
Theorem 13.2. The integral variables
x and yif \
(13.6)
and only
> 0, c +c 2
real positive in
> 0, then
variables
(13.1) converges
the
3
x and y. for
real
positive
if
2sgn(c +c . ) + sgn(r +y -1) > 0,
for
k,j =1,2,3 and k * j;
d i n <-|sgn(c + c )+sgn(c + c ), sgn(c +c ) + sgn(c +c ), 2min
sgn(c + c ) + sgn(c + c ) > + sgn(^r + y + ^ - 2) > 0 .
Now we assume that the contours L L
c
= {Reis)
= c> and L = L^ = {Reit) 2
and L 1
are vertical lines L = 2
l
= C}. Then, any reduction formulas
C
in §12 for the H-function (10.1) will similarly define the relations for the G-function
(13.1). For example, from
(12.3),
(12.4) we obtain the
sum and maximum properties for the G-function (13.1) of two variables as follows:
m,n:1,0;1,0 (13.7)
p , q : 0 , l ; 0 , l x,y
(a ): —
; —
p
(0 ):
0
0
^m+1, n ' p >q + i x+y
q
m,n:1,0;1,0 p,q:l,l;l,l x,y
(a )
0, o ) q
(a ): 1 ; 1 ) p
(0 ): 0 ; 0 q
(13.8) (a ), 1 m+1 ,n = GJ max(x,y) p+l,q+l
(x,y > 0 ) .
0, (0 )J q
91
y
Since for Y.L.Luke
the Meijer G-function (1969), O.I.Marichev
there are many particular
cases
in
(1983), A.P.Prudnikov et al.(1989), then
the two last reduction formulas give the corresponding particular cases for the G-function (13.1) of two variables.
For instance, from
_1,Of I - 1
G
x
o,i[ | 0 J
-x
=e
•
0,0:1,0;1,0 we obtain
0,0:0,1;0,1 x,y
—
: 0 ; 0
and
-
1,0:1,0;1,0
:1 ;1 1 = G 2'°(max(x,y)| o ^
0, 1: 1, 1; 1,x,y 1
1
]
: 0 ; 0
= exp[-max(x,y)]. The last equality can be written as follows r(i+s+t) -s -t , ,. x y dsdt =
1
{ZniV L
where Re is)
, ,, r exp[-max(x,y) ] ,
CLc
= c > 0, Re{t)
= C > 0.
By applying Theorem 7.3 on the double Mel1 in transform from the general H-function we obtain 00 00
3Tt Jexp[-max(x,y) ] ; s,t
r(l+s+t) st
exp[-max(x,y)] x
Re{s)t
Reit)
y
dsdt
> 0.
Consequently, we get the accordance of the double Mel1 in transform and its inversion 92
> r ( 1 ^ + t ) ,,
exp[-max(x,y)] <
Reis)
Rei))
> 0,
> 0.
For convenience here we construct the table of particular cases of the G-function (13.1) of two variables of the last relation type. Further in all the next formulas Reis) the
integral
exponential
Viv,x) bility's integral; Viv yx)
function;
Re i)) > 0, Rei))
erfc(x)
> 0 and Ei(x) is >
is the additional
proba-
is the non-complete gamma-function; J (x) is
¥(a,b;x) is the the Bessel function; K (x) is the Macdonald function; *(a,b;x) degenerate
Tricomi
function
(see
A.Erdelyi
et
al.
(1953)
and
A.P.Prudnikov et al. (1989)).
TABLE 13.1
h*(s,t) = |h(x,y)x h(x,y)xs-1yt"1dxdy
h(x,y)
II
e" x " y
rfs.tl, rls.tl, Re(s) > 0, Re(t) > 00
2
exp[-max(x,y)l exp["-max(x,y)l
-- iL ^- rfl+s+tl, iTl+s+tl , Re(s) > 0, Re(t) > 0
3
exp[-xV-xV] expf-xV-xV]
1 1
e-x-y
1
_i_
|A|
r[
1" Bs-At
5^tA
L
-bs+at "1
, =^pt A
'
],
J'
A = aB-bA * 0
( A ) j (B-b)s+(a-A)t
4
B
exp[-max(xV,xV)l exp [-maxfxV y )l
^
(Bs-At)(at-bs) (Bs"At)(at"bs)
x r r(B-b)s+(a-A)t [(B-b)s+(a-A)t "1]
93
5
(l+x+y)""
6
[l+max(x,y)l
a-s-t I
£3,t
r
a
J
(1-x-y)""1
8
(l-max^y)]""1
g
1 1-x-y
1u
r
r
>
Re{s+t)
s, t,a Re(a)
_ a+s+t
> 0
1" 1+s+t
r(a) . L Reioi) > 0 "it" l L a+s+t
' s,t,l-s-t
r
1 l-max(x,y)
"1
— +s+t, — 2
1
-s-t
, Reis+t)
J
2
1+s+t,1-s-t
rf
— +s+t, — - s - t
11
Re{s+t)
a
L
1n
>
' 1+s+t,a-s-t-I
i
Reioc)
7
, Re(oi)
2
< 1
s,t,p--s-t -s-t
r
-P |1-x-y|
2
Re(s+t)
'
J
;+t,
L
— £
_U+p)__
s-t o L
7T r(p)C0s(p7T/2)
Re?(s+t
94
< Re(p ) < 1
<1
12
1
-f> |l-max(x,y)|
"it
r
r[
s,t,p-s-t
(1
" P ) +s+t,
(1+P)
2
-st _
2
n (p)C0S(p7T/2)
Ke(s+t) < /te(p) < 1
13
Ei(-x-y)
14
Ei["-max(x,y)
"
r(s)r(t) sTt
r(s+t) sTE
" s,t, - +s+t 1 15
erfc(vS^)
1+s+t
J
r( 1 +s+t) erfc(^max(x,y))
16
V^
1 7 i /
1R 1o
19
l0g
(
1
1 na( 1 1 + x+y
- r[s,tf-s-t|, / t e ( s + t ) < 1
)
- -^-r-r[l+s+t,-s-tl
log 6 1+ [ max(x,>
1
St
, Ke(s+t) < 1
1" S,t,S +t,l-S-t
1 I
n r
^h-^l
- +s+t,l+s+t,i -s-t L
2
2
/te(s+t) < 1 95
»
20
log 1-
s+t,1-s-t
1 max(x.y)'
st
l , . l . - +s+t, - -s-t L
2
2
/te(s+t) < 1
21
log(x+y) x+y-1
4s,t,s+t,1-s-t,1-s-tj,
Re(s+t)
22
- i ^ r|s+t,1+s+t,1-s-t,1-s-t],
log[max(x,y)3 max(x,y)-1
Re{s+t)
23
s,t,s+t,1-s-t,1-s-t
log(x+y) x+y+1
-n r - +s+t,
- -s-t
2
2
Re(s+t) < 1
s+t,1+s+t,1-s-t,1-s-t 24
log[max(x,y)3 max(x,y)+l
-7T r st
- +s+t,
- -s-t
2
2
s,t,s+t,1-s-t 25
n r
log 1x+y
- +s+t,1+s+t, - -s-t 2
'2
Ke(s+t) < 1 s+t,1-s-t 26
log 1-
max(x,y)'
st
- +s+t,
- -s-t
2
2
Re(s+t) < 1
27
log(x+y) x+y-1
]■
T|s,t,s+t,1-s-t,1-s-t Re(s+t) < 1 96
28
H>
-~ rTs+t.i+s+t.i-s-t,i-s
log[max(x,y)] max(x,y)-l
Ke(s+t) < 1
29
log(x+y) x+y+1
-TT
s , t , s + t , l - s - t , l - s - t "I
r
i +s+t,
i -s-t
Reis+t)
J
>
< 1
s+t,1+s+t,1-s-t,1-s-t 30
log[max(x,y)] max(x,y)+l
r
-7T
"it
i +s+t,
\
>
-s-t
/te(s+t) < 1
31
r s,t,s+t+ - , - -s-t
log l+v^+7
2
7i r 1
[
s+t,s+t+l,
1
2
1-s-t
|Re(s+t)| < i
32
log
r s+t+ -
l+AAnax(x,y) 1-Vmax(x,y)
"it
, ~ -s-t 1
2
r
|_
s+t, 1-s-t
J
'
|Re(s+t)| < i
33
34
lo v ^
fs , t , s + t , s + t +
I
+ y^l+x+>
—n
ZVxl
l o / m a x ( x > y ) + v'l+maxCx.y) 2vW(x,y)
i , 1 - s - t "I
1+s+t, 1+s+t
1 2v^st
[
1+s+t, 1+s+t Re{s+t) < 1
97
J
'
'
35
1
2 l+/lTx~+7
s,t,s+t,s+t,l-s-t "
1rf
> s+t+ i
,s+t+1 J
2
Re(s+t)
36
lo
^
2 l+v'l+maxtx,y)
r
"s+t,s+t,l-s-t
r
lit
vW(x,y)
< 1
s+t+
'
-
L
J
2
Re(s+t) < 1
37
1
|r
d I U 5D 111
•l+x+y
"s,t,s+t + -
, -
2
2
-s-t
S+t+ 1 Re(s+t) < -
38
1
3 r p ^ l v\
^
^l+max(x,y)
r|s+t+ i
, i -s-tl
Re(s+t)
39
< i 2
rs,t,s+t+y ■
1 s+t
r(p,x+y)
,
r r
> L
s+t Re(s+t+v) > 0
40
r[i>,max(x,y)]
41
J ^ )
-4 Hs+t+i/), st
r
Re(s+t+i>)
" S,t,S+t+
> 0
-£>
_ s+t, 1+-^--s-t _ --Re ( — )
98
42
J
(2v'max(x,y)]
s+t
"it
r c+t+ J^_
r
44
45
sin v ^
sT
vW(x,y)
r
v^ r
cos(2v6^)
2
2
Vtt
cos(2v'max(x,y))
it
[ S,t
KV( H
2
1
- "S-t J 2
, Re(s+t) < 2
r i+s+t r
irf 2
2
, Re(s+t) < -
L s+t
1
, Re(s+t) < 2 - -s-t 2
s,t,s+t+ — , s + t 2
L
2
>
S+t
Re(s+t) >
48
Re(s+t) < -
r - -s-t i
2
L
47
— )
_ s+t,l+s+t
L
46
J
's,t, I -s-t -1 2 , i/£ r
2
sin
»
1+ — -s-t 2 -Re(
43
i
J
\Re{v)\/2
I^r[s+t+^,s+t- ill,
K^(2^max(x,y)J
Z St
1
2
2J
Re(s +t) > \Re{v )|/2
99
F[;H
49 i
i
L
'
|s,t,a-s-t]
■[.b]
b-s-t
j
Re (s+t) < /te(a); b * 0, -1 , - 2 ,
l+s+t,a-s-t 50
P * -max(x.y) i iL I J
b-s-t Reist)
b * 0,-1
,-2,...
s,t,a-s-t,b-s-t
■[.%]
■ [v H
51
< Reia);
1 st '
c-s-t Reis+t)
< Re(a),
Reib);
c * 0,-1,-2,...
52
"
'
l+s+t,a-s-t,b-s-t
■[.%]
-max(x,y)
c-s-t
1 st
Rei s+t) < /te(a),/te(b); c *
0,-1,-2,
^s,t,s+t+l-b,a-s-t 55
* a,b;x+y
a,
a-b+1
Reib)-1
< /te(s+t) < fle(a)
l+s+t,s+t+l-b,a-s-t] 56
tf a,b;max(x,y)
st
a, Reib)-1
100
a-b+1 < Reis+t)
< Rein)
A:B:B' f (a):(b);(b>
(c),(d),(d')
57
-x>-y C:D;D'
(c):(d);(d');
(a),(b),(b')
(a)-s-t,(b)-s,(b')-t,s,t x r (c)-s-t, (d)-s, (d')-t
Finally, we give a representation of the G-function
(13.1) through
the sum of double hypergeometric Kampe de Feriet series (1.7) in the case m
= 0 (see also formula (14.1)) ,
*1
M
l
m
*2
M
2 '*3
M
,
(2).
): (a
f
(3) .
); (a
C0 (1) ): (Hl2));
3
A
) )
(P<3))
l+P(2V3)-Caa))
m
k k=l
(1),
(a
0 ,n :m ,n ;m ,n 1 2 2 3 3 x,y p ,q : p ,q ;p ,q
j
(a'")
j=l
1,n 2)
-p' -p! 3, > l +/ 3' 2, + p! 3) -(3 <1) ) i
,
n + l , p k
j
k
(2) (0 )*l,m - Uk( 2 , l l V k2 , - ( a l 2 ) ) l,n ' 2
X r (2)
2
(<x )
j
l,q
2
2
-p' >, 1 + p <2>-(^ >)
n +1, p 2 2
k
k
m +1 , q ■ 2 2
(13.9) (0 ( 3 ) )* -^(3),l^(3)-(a(3)) 1,«3 J J l>n3
X r
(a(3)) ^^3) n3+ l,P3 j
P : p
;P
xF q :q -l;q -l M M M
l
2
3
(3). (3), j V1>q3
l,p
'k 1
j
l,p
k 2
1-(P11,)1l,q + 8 k< 2 V 3j) : 1-(P<2')*l,q V 2 k' ; 1
2
101
l-(a ( 3 ) )
V3'; I,P3
J
p +p +n +n +m 2
(-1) '
*
2
p +p +n +n +m
2
x,
(-1) *
3
3
'
3
I-(P(3)); v . 3 , ; 1 , q
(a)
3
J
(p-q) is the abbreviation of the array of q-p+1 parameters p,q
(a)
(a)
= a ,a p
, ..., a ;
p+i
q
-a (l^k^p) denotes the array of p-1 parameters
1, p
k
(a)
-a = a , . . . , a k
i,p
and
p,q
1]
T The
l
,a
k-i
k+i
,...,a; p
is Slater's notation (4.33).
formula
(13.9)
holds
provided
the two following
sets of
conditions satisfy
(13.10)
p + p < q +q ,
p +p
3 ^1 3
*1
|arg(x)| < ^ (2^+ 2m^ \arg{y)\ J &
(13.11)
2 ^ - p ^ p ^ ^ ) ,
< ^ (2n + 2m + 2n -p -p -q -q ), n 2
1
3
3 *i *3 n
3
|arg(x)-arg(y) | < ^ (2m + 2m +2n +2n -p -p -q -q ). &
Here
(13.10) provides
&
2
2
the convergence
3
2
3 *2 *3 n 2
n
3
of the double Kampe de Feriet
series, which are involved in (13.9). The conditions
(13.11) give the
existence of G-function of two variables in the case ml = 0. We conclude by remarking that the representation (13.9) was obtained by R.P.Agarwal (1965) and B.L.Sharma the
last
inequality
corresponding
double
(1965), however, in both works there was omitted in (13.11). integral
As is known from
(13.1)
which
G-function of two variables can be divergent.
102
is used
Theorem
13.1,the
to define the
§14. The Double Kampe de Feriet hypergeometric series
J.Kampe de Feriet
(1921) initiated the study of the case B = B'and
D = D'of the double hypergeometric series
A:B;B' ( (a): (b);(b'); (14.1)
x,y (c):(d);(d');
C:D;D'
A T1 T
r
'
(a )
n
j m+n ' '
J
i - 1
C n=0 ~TT(c ) 1 ' j=l
B' B ~TT (b ) T~T (b')
j m+n
5 — 1
D FT
' ' j=l
j m ' '
J
in
X
m
n
y
\ = 1
D' (d ) ~TT (d') j m
' ' j=l
m! n!
j n
where (a) denotes the Pochhammer symbol defined by n
1,
if n = 0,
(a) = a(a+l). . . (a+n-1),
if n € IN = {1,2,3
>.
For the above double series H.M.Srivastava and M.C.Daoust (1972) deduced from much more general results (proved by them) that (i)
if A+B > C+D+1 and
A+B' > C+D'+l,
then
the
series
(14.1) diverges whenever x * 0 and y * 0; (ii)
if A+B = C+D+1 and
A+B' =
C+D'+l,
then
the
series
(14.1) converges absolutely, provided that
( (14.2)
max{Ixl,lyl> < 1 when A ^ C, i l/U-C)
|y| 1 / ( A _ C ) < 1 when A > C;
(iii) if A+B < C+D+1 and A+B'< C+D'+l, then the series (14.1) converges absolutely for all x,y € C. 103
It is understood (in each
situation)
that
no zeros
appear
in the
denominator of (14.1). In case (ii) above, the double series (14.1) also converges when x and y lie on the boundaries of the regions described by (14.2) provided some additional constraints, which were found recently by Nguyen Thanh Hai, O.I.Marichev and H.M.Srivastava
(1992) are imposed also Nguyen
Thanh Hai (1990a)). Our
solutions
of
the
convergence
problem
for
the
double
hypergeometric series (14.1) are contained in Theorems 14.1, 14.2 and 14.3 bellow. Theorem 14.1. Let A+B = C+D+l, A+B' = C+D'+l, and A = C. Then
the
series A:B+1;B'+1 f (a):(b);(b'); (14.3)
x.y A: B ; B*
(i)
converges
[ (c): (d); (d' );
absolutely A
when |x| = 1 and |y| = 1, if and only B+l
A
B
j=i
j=i
A = Re
if
< o, L
j=i
A
j=i
B'+l
B'
8 = Re
< o, j=i
j=i
j=i
j=i
and
A
B+l
j=i
j=i
B'+l
B
B*
j=i
j=i
c = Re
(ii)
< o,
converges
conditionally
if
A < 1, 5 < 1,
j=i
j=i
when |x| = 1 and |y|=l (x * 1; y * 1), and 104
c < 2;
(Hi)
diverges
when |x| = 1 and |y| = 1, if
at
least
one of
the
following three conditions does not hold true: and
A < 1, 5 < 1
c < 2.
Theorem 14.2. Let A+B = C+D+l, A+B' = C+D'+l, and C-A = k > 0. Then the
series A : B+k+l;B'+k+l f (a):(b);(b'); x,y A+k: B ; B' (c):(d);(d');
(14.4)
(i) converges absolutely when |x| =1 and |y| = 1, if and only if A
B+k+1
j=i
j=i
A+k
B
A = Re
< o, j=i
j=i
and A
+
B'+k+l
B'
c
& I ; -Z r &
5 = Re
j=i
(ii)
A+k
b
j=i
converges
conditionally
if
A < 1,
j=i
< 0;
j=i
when |x| = 1 and | y | = l (x * 1; y * 1),
and 5 < 1.
Theorem 14.3. Let A+B = C+D+l, A+B' = C+D'+l, and A-C = k > 0. Then the
series A+k:B+l;B'+l ( (a):(b);(b') ;
(14.5)
x,y A :B+k;B'+k
(c):(d);(d');
converges absolutely when |*|1/k
(14.6)
+
|y| 1/k = 1
(x * 0; y * 0)
if A+k (14.7)
e = Re
B+l
B'+l
A
B+k
B'+k
& 1 v I ;- I i - 1 v I d ; j=i
+
j=i
b
j=i
j=i
105
c
j=i
j=i
+k < 1.
Proof of Theorem 14.1. Case A
m
(i).
Denoting
the
general
term
of
the
series
(14.3)
n
x y , and making use of the familiar asymptotic estimate:
T(a+n)
(14.8)
Re(a-b)
,
.
we have
A FT 1
l A
I J=l A
m nI
A x y
=
B+l B'+l F T (b ) F T (b')
(a )
j m+n ' ' J=l B
"pj(c) 1
' j=l
j m ' ' j n J=l B'
m X
n y
7 7 (d.) 7 7 (<*•.)
j m+n
' ' j=l
(C ),(d ),(d') J J J
j m
' ' J=l
j n
a +m+n,...., a +m+n 1
|x| m |y| n
(a ),(b ),(b') J J J
A
, c +m+n, L
c +m+n
l
A
(14.9) b +m,....,b +m,b 1
B
+m
.,b',+n,b', +n B B+l
d'+n,..
.,d',+n,1+n
l
d +m,.....d +m,1+m 1
b'+n,.,
B+l
l
B
, .a /3-i 3T-i H (m+n) m n
B
TT f
where H
is a constant,
(m
> co; n
|x| = |y| = 1,
B
(14.10)
a =
Re L
D
B'
D'
j=l
j=l
Y*rYc\,v--Re\Y*i-Y«3[*--"e\Y?]- Id; j=i
j=l
j=i
j=l
J
L
C,D,and D'(and A,B, and B') being always specified in the context, and
r(a )...T(a ) i
b , . . . J,b
• l
p
is Slater's notation.
106
p_
r(b )...r(b ) i
p
by
To prove Case (i) of Theorem 14.1, it is sufficient to apply the following result.
Lemma 14.1. Let
a,/3,y e IR. Then
I
(14.11)
)
the
series
(m+n^'V""1
m,n=l converges
if
and only
if
a+|3 < 0, a+y < 0, and a+ft+y < 0.
Proof. Let us fix m = 1 (n = 1) and obtain the condition a+y < 0 (a+/3 < 0 ) . Hence 2a+/3+? < 0; and if a ^ 0, then *+($+? < 0.
In the case when a < 0, we have
(14.12)
>
(m+n) m
n
)
=
m,n=l
(m+n) m
+ )
n
m>:n^l
(m+n) m
n
,
l^m
and
(m+n) m'
n
=
m
n
1+ -
.
I mJ Since m ^ n, the hypothesis a < 0 implies that
a 2
Now
if we denote
"
£
(
1+
i)
the simultaneous
< 1.
convergence
or divergence
of
two
series by the equivalence symbol =, then the following relations are easily verified: co
)
(14.13)
i>n^l
mi
oo
im+nj m
n
=
>
oo
(.m+nj
rn^n^l
n
-
)
m
,
m
m=l ^
107
\
> n n=l
'
( lPct+0+y-i i^m"^"1 m=l
I
(y ( y > 00)).,
oo
« * JI
Vm^^logdn) V ma+/ log(n) m=l m=l oo a+8-i °° m
~ 1 I I I
E
I
[ m a+ ^ _1
(y (y == 00), ),
(y < 0 ) . (y < 0 ) .
m=i
Hence the condition a+/3+y < 0 follows, and the proof of the necessity of Lemma 14.1 is completed.
Next we observe that if a £ 0, then mmaa**fifi'1'n1n7r7r**0La""11..
3r_1 a Cm+n)am^"1n3r ^^
Thus the convergence of the series (14.11) follows from the conditions a+£ < 0 and a+y < 0. On the other hand, if a < 0, then the convergence of the first series on the right-hand side of (14.12) follows from the relationship
(14.13).
The
convergence
of
the
second
right-hand side of (14.12) can be proved similarly.
Case(ii). Casedi).
on
the
We shall need the following result in this case.
Lemma 14.2. Let
a,/3,y e R, and = (m+iO^-n-1.
e
Then, for Then, for
series
■
the the m,
equalities: equalities:
lim e n—Xx> lim e
m, n — X »
to hold true,
mn mn
it
= lim lim c = m—>oo lim n—>oo lim c m—X»
is necessary
= lim lim e m—X» e = n—>oo lim lim
n—X»
n—X»
and sufficient
m—Xx>
that
en en
=0 =0
a+|3+y < 2, a+/3 < 1, and a+y < 1. a+^+y The proof of Lemma 14.2 is easy, and we omit the details involved.
Setting A x m y n = u v mn
mn mn
, where v mn
108
=
x m y n and
A
(14.14)
u
j[= 1 '
-
j
7 7 (b.) 7 7 (b'j m+n j =', 1 '
A
j mJ =[ 1 '
B '
j
l e t us use the conditions
j
1 n
B'
1
m!
n!
7 7 (d ) 7 7 (d'j
77(c) 1
B'+l
B+l
7 7 (a.)
m+n
'
'
|x|=|y|
j m '
=1
'
j
n
(x * 1; y * 1 ) ,
a+0 < 1, a+y < 1, and a+/3+r < 2 . Then i t
i s r e a d i l y seen
that in
1. The p a r t i a l
2.
{u
3.
the
sums S
mn
=
ii
)
)
L
v
L
ij
a r e bounded;
> c o n v e r g e s u n i f o r m l y t o z e r o when m —> oo and n —> oo; mn
series f |u
L ' m,0 m=0
- u
I,
m+1,01
f
lu
L ' 0,n n=0
- u
I
0,n+l',
oo
and
) | u - u
U m, n = 0
' mn
m+l,n
- u
m,n+l
+u
I
m+l,n+l '
are convergent. By appealing now to Theorem 1.1.3 of A.Yanushauskas the conditional convergence of the series
(1980), we obtain
(14.3) under the specified
conditions. Case
(iii).
It
is
obvious
that
the
condition
mn m, n — > oo
necessary for the convergence of the double sequence {u
>. Therefore,
mn
the validity of the assertion of Theorem 14.1 in Case (iii) follows from the relationship (14.9) and Lemma 14.2.
■
Proof of Theorem 14.2.
Case
(i).
With the help of the asymptotic estimate (14.8), we find
for the general term of the series (14.4) that
109
,I
A
1 1 1 1
(14.15)
j j m m ' ' ' '
—— B
—— B'(d B'
j j n n
X X
rr«v.-» r r «v.rT (d ; }} » rr«v.-» r r «v.rT ; »
1
j=
i
J=
I
j=
yy
m! n! m!
i
.a+k 6-k-i y-k-i ii iimm i i (( nf n f m!n! m!n! 11 TT , r, ~ H (m+n) m nn xx yy r, where H
. -
B'+k+l
T T (b.) (M T FT <»f>.)•.) T~f 7 (b
j j m+n m+n ' ' ''
|B |B x"y"| x"y"| == MM^^ mn mn A+k A+k
(14.15)
B+k+1
T7 7 (a.)
is a constant, and a,0,* are defined
,, .. Cm —> °°; oo; n —> °°)» oo), (m
by
(14.10) with
the
2
numbers of parameters appropriately specified as in (14.15).
numbers of parameters a p p r o p r i a t e l y s p e c i f i e d as in (14.15). By Stirling's formula:
By S t i r l i n g ' s formula:
. . ,,
we have we have
, , , , ,— —r.~ m!n!
/=— .. k+1/2 k+l/2 pr-
k! k! ~ ~ v27i v27i k k
e e
m+l/2 m+l/2
n+1/2 n+1/2
V^TT pr- m
-k -k
,...
,,
(k (k —> —> oo), oo), (m
oo;
_>
n
oo).
_>
(m —> co; n —> oo).
n
2 (m+n)! ((m+n) m+n)m+n+1/— ( i ^! Then, for Ixl = 1 and lyl = 1, we find from (14.15) that Then, for |x| = 1 and |y| = 1, we find from (14.15) that j-
,„ ,„ |B
m
n
-i
K
m n, ,_ .kTT . .a+k/2 /3-k/2-l y-k/2-1 j-— m n -i k m n, ,_ .kHTT (m+n) . .a+k/2m /3-k/2-ln y-k/2-1 — xy | ~ (271) (m+n) m + n mn 2
|B xy R| H ~B' (271) H2(m+n) = mn(2TT) .
m
(m+n) m + n
n
2 mn
= (2rr)kH B' . Furthermore, for sufficiently large m and n, 2 mn Furthermore, for sufficiently large m and n, ro n n m m n
, ,
= =
.m+n .m+n
'•*"'
Hence Hence we we obtain obtain
1
I-
1
//
y,m ^m
f,
^n xn
M) M) B' B'
(14.16)
..
\-
mn mn
< <
1
<< _L _L — —
(p R+). (p €€ R+).
pp p p
" ■
a+k/2 1 2r (m+n)a+k/2 m^"k/2"1_ "pkn "k/2"11~pk "pk. (m+n) m^~k/2~ pkn 2r ~k/2~
In accordance with Lemma 14.1, the double series with a general term of the type (14.16) converges for sufficiently large values of p. So, by taking
into
account
the
conditions
conclude that the series 110
A = a+/3 < 0
and
S = a+r < 0, 6
we
I
m,n=l converges. The sufficiency of the conditions in Case (i) of Theorem 14.2 is proved.
The necessity will become evident if we fix m = 0 (n = 0) in the series (14.4).
Case
(ii) of Theorem
Theorem 14.1.
14.2 can be proved
in the same way as in
■
Proof of Theorem 14.3. Let us consider the general term of the series (14.5) (m+n)! , „ .
. „ .
(14.17)
i _ 1
m Hi
C x y mn
TT
'
= HC
/-
xtt+k
~ H (m+n) 3
/3+k-l
3f+k-l
na
m'
i
i
x
m
i
i
n
y
xT y ".
3 mn1
' ' '
By the principle of mathematical induction, it is not difficult to obtain the inequality: -.k (m+n)!
(14.18)
[k(m+n)]! (km)!(kn)!
Now c h o o s e M € IN so
that
kM+/3+k-l > 0 Then we g e t t h e
and
kM+y+k-1 > 0.
inequality: kM + 8 + k - l
kM+^+k-l
m n 2kM+2k + |3+3r-2 (m+n)
Hence
111
< 1.
y c'mn x
I m I
in
y
zL
i,n=l 00 r—«t
kM k -- Il K P l ++ p/ 3++K
_
) (m+n) / *—■ „ m,n=l
,
(m+n)!
m n
I )
.2kM+2k + j
(m+n)
m+n)
<
(m+n)
x
y
(m!n!)
v 2M
a+/3+3-+k-2
(m+n)
m n
m,n=l nn,n= 00
(14.19)
^kM+^+k-1
-I
(m+n)
*
-l
(m+n)!
(m+n+2M)!
a+p+^+k-2
x
*
y
(m! n! )
|x|m|y|n
(m+M)!(n+M)!
m,n=l
| xy | '"
a+|3+3r+k-2 (m+n)
>
m,n=l
(
[k(m+n+2M) ] ! [k(m+M)]![k(n+M)]!
ky K/
xk(M+m) N mn+nu
k, •, K/
Nk(M+n) xH
00
xy|-"^l-^*k-a(^
k>
+
xk(2M+l)
{^f)'
1=2 00
=
xy
M^
i a + / 3 + r+ k-2
js.nce ^
+
^
=^
The last series converges if a+/3+r+k < 1. This evidently completes the proof of Theorem 14.3.
■
For an illustration of the above Theorems 14.1 - 14.3 we consider the well-known Appell series F
- F
(see A.Erdelyi at al.(1953), V.l)
which are special cases of the Kampe de Feriet series (14.1). 112
Consequence 14.1. The first
Appell 1:1; 1
(14.20)
F (a.p.lTjrjx.y) =
series
f a:0;/T;
F
x,y 1:0;0
y:-
(a)m+n ^ O )m (£') n
C O converges absolutely
when |x| = |y| = 1, if and oniy if
< 0,
Re(a+^'-r) < 0
(iij converges conditionally
diverges
when
|x| = 1 and
following
three
conditions
Re{oc+f5-v) * 1,
and Ke(a+/3+0* -y) < 0;
when |x| = |y| = 1 (x*l, y*l), if Re{oc+(S*-y)
Ke(a+/3-r) < 1, (Hi)
y
(y) m+n
m,n=0
Re{a+p-v)
x
< 1
and
Re(a+^+j3' -y) < 2;
|y| = 1, if
does not hold
Ke(a+£'-y) * 1
and
at
least
one
of
the
true: Re(a+/3+/3' -y) < 2;
Remark 14.1. In the literature, see, for example, P.Appell, J. Kampe de Feriet
(1926), A.Erdelyi et al. (1953,v.l), there is the following
formula (14.21)
Fta.p^'jy;!,!) = T r-a, y-/3-/T
(14.22)
However,
/te(a+0+0'-r) < 0.
provided
the
condition
(14.22)
is
given
the
series
F (a,j3,j3* ;y; 1,1) is not obligatory convergent, i.e., the limit N
lim M , N — > co
does not always exist
N
II-
^ ^ J . «
m ! n !
m+n
m=0 n=0 (for example, when a = y, £ = 1/2, £*= -3/2).
Therefore the formula (14.21) means 113
(a) A (0) (0») m+n m n
F (a,/3,/3' ;y;l,l) = lim k—xx)
( y m+n )^
m+n=k
m!n!
k y,y-a-/3-/3' lim k—x»
y-a, y-/3-/3' J
j=0
provided (14.22) holds true.
Consequence 14.2. The third
Appell
series
0:2;2 f -:a,a' ; 0;0' ;
=F
F (a,a;/3,/r ;y;x,y)
x,y 1:0:0
y:
(a) (a') (3) O ' ) m n m n
x m yn m! n!
Cr). m,n=0
(i)
converges
absolutely
when
|x| = |y| = 1, if
and only
if
fie(a+0-y) < 0 and Re(a'+ 0'-y) < 0; (ii)
converges
conditionally
when |x| = |y| = 1 (x * 1, y * 1)
Re(a+0-y) < 1 and Re(a'+ 0'-y) < 1.
Consequence 14.3. The fourth
Appell
series
2:0:0 f
a,0:-; -;
0:1;1
-
F (a,0;y,y' ;x,y) = f
x,y
n m,n=0
converges
absolutely
:y;y
(a) x (/3) ^ m+n m+n
x m y11 ^
(y) (y ) m n
m! n!
when v|x| + v|y| = l,(x * 0; y * 0) if 114
/teCa+p-r-y') < -l. Consequence 14.4. The second Appell 1: 1; 1 F (a,|3;/3' ;y,y' ;x,y) = P 0:1;1
series
a :0;0';
1 x,y
(14.23) (a) _,_ (0) O ' ) m+n m n
x m y11
(y) (y') m n
m! n!
_ m,n=0 converges absolutely
when |x|+|y| = l,(x * 0; y * 0), if
(14.24)
Re(a+0+0'-*-*' ) < 0.
Here Consequences from Theorem
14.1, 14.2 and 14.3 - 14.4 follow
14.1, 14.2 and 14.3 respectively.
immediately
It is interesting to
remark that (14.24) is necessary and sufficient for the convergence of the series F
(14.23).
Theorem 14.4. For the second Appell following
three statements
1) The series
that
series
(14.23) the
equivalent
(14.23) converges
y = 1-a, where 2) The series
are
absolutely
at
least
for
x = a,
a € R, 0 < a < 1.
(14.23) converges
absolutely
for
all
x,y e C such
|x| + |y| =s 1, x * 0, y * 0.
3) The inequality
(14.24) holds
true.
Proof. First note that the chain 3) =» 2) => 1) is evident. We must prove that the statement 3) follows from 1). Since for the series F ^
2
(14.23) the following symmetry property holds valid
F (a,|3;0';y,y';x,y) = F (a,0f;0;y*,y ;y,x),
115
then we can consider 1/2 ^ a < 1 without emissing a community. In this case for the second Appell series F
we write the asymptotic representation
obtained by M.Saigo, O.I.Marichev and Nguyen Thanh Hai (1989)
F (<x,/3;/3' ;y,y' ;a-p,l-a-5) ( a, l + a - y , y ' -/3'
y,y-a-/3 F y-a,y-/3_
J 3 "^
3 2[
l+a+0-y,
y'
a-1 a
y , y ; a + / 3 - y , y + y ' -a-|3 -a)y"a^?
r
a , / 3 , y ' - / 3 ' , y+y' -a-0
(14.25) i-/3,y-/3,y+y'-a-/3-/3
a-1 a
x F 3 2
1 -a-/3+y, y+y' -a-/3
y,y', a+/3+/3'-y-y' (p+5)y+y -a-
a ^ ( 1 _ a ) /3'-y' +
Q(]
a,/3,y
where p F
+0, 5 ^ + 0 , y-a-/3, y+y'-a-/3-/3' are not equal to integers, and
is a Clausenian function (A.Erdelyi et al. (1953), V.l).
3 2
Now let in (14.25) p and 5 —> +0. Then by second Abel's theorem (see, for example, A.Yanushauskas (1980)) the left part of (14.25) tends to a finite
limit F (a,/3;/3'; y, y'; a, 1-a) since
(from
the statement
1 this
series converges absolutely for the point (a,1-a)). On the other hand 'x+'x' — oc—fi- 8'
the right part of (14.25) contains the expression (p+5)
' ' , that
is why it converges to a finite limit if and only if Ke(<x+/3+£' -y-y' ) < 0. The proof of Theorem 14.4 is completed by following two notes. At first, here the conditions y-a-/3, y+y'-a-/3-/3' = 0,-1,-2,
are
not essential.
At second, in the case a = 1/2 the formula is also true, if we set 116
(a,b,c; F
fa,b,c;
) -1
3 2
lim
F , 3 2
d,e;
d,e:
z -^-1 Since the Clausenian function
F
is analytic near the point z
limit always exists (although the Clausenian series
-1, this
F (z) converges for
z = -1 if and only if Re(a+b+c-d-e) < - 1 ) . Finally, we note that the Kampe de Feriet series can be represented by the G-function (13.1) of two variables as follows
A:B;B' f (a):(b);(b'); (14.26)
p
(c),(d),(d>) -x,-y
C:D;D'
(c): (d);(d');
(a),(b),(b')
(27Ti)
(a)-s-t, (b)-s, (b' )-t,s,t' x y dsdt (c)-s-t,(d)-s,(d')-t L L t s
(c),(d),(d')
0,A: 1,B ; 1,B'
(a),(b),(b')
A,C:B,D+1;B',D'+1
l-(a): l-(b);
l-(b')
'
x,y l-(c):0,l-(d); 0, l-(d')
where (14.27)
A+B < C+D+l,
A+B' < C+D'+l
and f |arg(x)| < ^ (A+B-C-D), (14.28)
Urg(y)| < \
(A+B'-C-D'),
Urg(x)-arg(y)| < | (B+B'+2-D-D').
(14.28a) (14.28b) (14.28c)
Here (14.27) provide the convergence of the Kampe de Feriet series (14.1), but conditions (14.28) are required for the convergence
117
of
the
double contour integral in (14.26). The relation (14.26) can immediately follow from the series representation of the G-function (13.9). We
conclude
by
remarking
that
provided
(14.28) holds
true
the )
A:B;B'f(a):(b);(b*); representation
(14.26) gives a meaning to f
-x,-y
C:D;D' [(c): (d); (cT); in the case when (14.27) does not hold true.
118
J
CHAPTER III. ONE-DIMENSIONAL H-TRANSFORM AND ITS COMPOSITION STRUCTURE §15. Spaces UJf^L) and 3JT1 (L)
In this Chapter we will consider a generalization of
the
next
integral Mellin type convolution transform
(15.1)
(Kf)(x)
k(xu)f(u)du,
x > 0,
o where k(x) in general case is Fox*s H-function of one variable (1.5). The transform (15.1) in various forms is considered by many mathemati cians
in
traditional
Mellin-Parseval
spaces
Lp (1 ^ p ^ +»). Here
formula we can represent
with
the transform
the
help
of
(15.1) in the
next type r+ioo
(15.2)
k (s)f (l-s)x"sds,
(Kf)(x) = ^ I r-iw
where f (s) denotes the Mellin transform of functions f(x) (see, for example, E.C.Titchmarch (1937), O.I.Marichev (1983))
(15.3)
f (s)
(y-ioo,y+im)
f(x)xs~ dx
is some vertical contour in the complex plane s. Thus the
transform (15.1) can be studied with the aid of asymptotic
estimations
of the functions k (s) and f (1-s) on the contour (y-ioo, y+ioo). It not
difficult
defined
on
to notice the
replacement
of
convolution
form
that
same
line.
variable
we
if
y = 1/2,
Putting can
write
as follows 119
then
these
f (x) = x f(x transform
functions )
with
(15.1)
in
is are
simple Mellin
(15.4)
1 2ni
k(-)f(u)-
(Kf^Hx) o
where
k (s)f (s)x ds,
= 1/2}. Mainly we will consider transforms of
type form (15.4) due to its large convenience for our further studies. Thus the behaviour of the functions k (s) and f (s) on the contour cr forces us to give the definition of the special functional space 3JT (L) introduced by Vu Kim Tuan, O.I.Marichev and S.B.Yakubovich (1986). As it is showed
below,
this
space
is very
convenient
for
the
studies of
transform (15.4).
Definition 15.1. Denote by 311" (L) the space of functions f(x), x € (0,+co),
representable
by
inverse
Mellin
transform
functions f (s) € L (o*) = L(tr) on the contour
(15.5)
The
f(x) = CTf1/ f*(s); x j = ^ j
space
3J1~ (L)
with
the
usual
of
€ C, Re(s)
integrable = 1/2}:
f (s)x ds.
operations
of
addition
and
multiplication by scalar is a linear vector space. If the norm in 5Jl~ (L) is introduced by the formula
(15.6)
|f (l/2+it)|dt,
m (D then the space 3H~ (L) is Banach.
Now we consider the main properties of the space 3JT (L).
1) f(x) € aif^L) if and only if x"1f(x"1) € UJT^L). This property exists as a result of the fact that the functions f(x) and x~ f(x~ ) are the inverse Mellin transforms of the functions f (s) and f (1-s) respectively, which simultaneously belong or not to
L((r). 1/2
-1
2) If fix)
€ UK
(L) then x
f(x) is bounded uniformly, continuous
120
on ((0, on 0 , +00) +oo) and and ffurthermore urthermore x x TW: _
__^___J...
r_ I I -
u c
,
ff(x) (x) =o o(l) ( l ) when x -> -» 0o0o a and nd x x-> - >0. 0. r_ _ .«__
■|-V"i*=» ■f_V-i*=»
R i p m a n n - 1 (aV^occniP
1 ^mma
f_..
F* C m a r cr hh 1937) 1 Q^7 1 E C Ti T i ttcrhh m
e JJJf'dJ J f ' d J then x1/2f(x)g(x) e L). 3) If f(x), g(x) e e ^VC(1{L). This follows from the fact that x1/2f(x)g(x) is the inverse Mellin transform of the function ^
f*(T)g*(s-T+l/2)dx
which belongs to L(
by Fubini theorem. Here g(x) = aOJf^g'Cs); n ^ j g ' C s ) ; xj.
00
4) Let f(x) e JJf'CL) J J f ' d J and x"1/2g(x) e L(R L(R+). Then |g(u)f(jS)*H [ g ( u ) f ( ^ belongs belongs + ). Then 0
t o nf'cu. toJJf'CL).
In fact by the property of the Mellin convolution this integral is the inverse Mellin transform of the function f*(s)g*(s) and since f*(s) € L(cr) and g*(s) belongs to space of essential bounded functions on
As is known the Mellin products
of gamma-functions
transform
of H-function
and according
is the ratio of
to asymptotic
of gamma-
function this ratio has power-exponential behavior on the contour o\ Therefore it is necessary to take into consideration this fact in the spaces of type 3Jf * (L). Definition 15.2. Let c,y e K be such that (15.7)
2sgn(c) + sgn(^) ^ 0.
Denote by JJf1 (L) the space of functions f(x), x e (0,+co), representable by the inverse Mellin transform (15.5), where f*(s)|s | 7en° ' I m S U L(cr). Note that f*(s)|s|V c | I / n s | e L(cr) if and only if f*(s)|s|V C |s' e L( 0, y € R, 121
or c = 0, y > 0, which is equivalent to (15.7). The space 3JT
(L) is Banach with norm
enc\lms\
(15.8)
,sV(s)ds|.
ffl (L) c,7
It is obvious that the space UH
(L) in case c = 0, y = 0 coincides
with the space 3H (L). The following theorem takes place.
Theorem 15.1. For the
set
of spaces
(L) the
3fl
following
statement
c,y
is
true (L) c m'1
UJf1
(15.9) if and oniy
(L)
if 2sgn(c -c) + sgn(y -y) >: 0.
(15.10)
Proof.
Suppose
that
inequality
(15.10)
holds
valid.
Then
from
(15.8) we have
l f l -a m
e
Tic | J O T S |
I
y
*
^,
s f (s)ds
i
(D f
(15.11)
TIC | JOTS |
e
f :£ C
7i(c-c ) I Ims |
y
Is *f (s)e
TIC
I Jms I
e
*
y # |s *f (s)dsl =
s
y-y
^sl
CIIfII 3R
(D
c ,y I I
where constant C is defined by the inequality (15.10)
(
TI(C-C
) \Ims\
e
y-y
|s|
I < +oo.
Now suppose that inclusion (15.9) holds. Then f(x) e W f 1 (L) c lfl'1 (L) c .r c,y -TIC |J/ns| -y -l-e # X a if f (s) = e I s| , e > 0. For this f(x) from (15.8) we 122
have .,
e
m
,
Tic Ims
|
-7tc \Ims\
y
1
s e
i
-y
s
i
-l-c
_, i
ds
(D 7r(c-c ) | Ims | e
Hence for all e > 0
Is
y-y - l - e d s l < +oo,
e > 0.
we get
2sgn(c -c) + sgn(9r -y+e)
> 0,
which is equivalent to (15.10). This completes the proof of Theorem 15.1. Remark 15.1. Note that 3JT1 c
and y
= y.
l
Then for yt
y , e, 1
1
(L) s 3JT1 (L) if and only if c = c
1»y1
c
J
c,v
€ (R, moreover e > 0, e
l
l
> 0, we have
the following inclusions 1 an"1 (D = m' (u 0,0
D
1 m' 0 ,C
( D D JJT1 C
(15.12) 3JT 1
D
e ,y +e
D im" 1
(L)
c +e ,y
l i
From t h e l a s t p r o p e r t i e s f o r 3)1
(D
,2T 1 1
(L).
l
(L) i t
i s not d i f f i c u l t
to
obtain
c,y
the following result. Consequence 15.1. If pairs
(c ty
) and
(c ,y m'1
c ,y
where
the pair
(c,y)
inequality ),
m as
1
c ,y
for
some
c,y
follows < c,
1
, if c
(c,r) =
2
< c,
2
(c ,min(3r ,3r )) 1
also
(L) s 3JT1 (L),
, if c
1
valid
then
(L) U
is defined
(15.7) holds
2
1
, if c = c . 1
2
To establish further properties of the space 3fl following auxiliary Lemmas. 123
(L) we need the
Lemma 15.1. Let all
k k
also
r ( \ 2|t| 3 eexp x p Rjs+tl+Rjsl+R^tl Rjs+t|+Rjs|+R 3 |t| . the vertical
linss iines with
1/2, i.e. , (cr x cr ) = j(s,t) e C 2 , /Res)) t e s ) ) = Re(t) Reit)
if
sup S(s,t) < +oo for sup
and only
(15.15)
the
real
parts
= 1/2 V. V Then Then
(s,t) e (cr x cr ) s
t
if [
(15.15)
and
r
c
(15.14)
numbers
k k
r r ~(s,t) = |s+t| l\s\ S(s,t)
(15.13) Let
R , r , k = 1,2,3, be real
2sgn(Rk+R.) + sgn(r sgn(r k+r.) +r.) < * 0, 0, r3
3
J r^
f
1
1]
Y \
k,j=l
v
J
{
k *j k,j=i
v
J
{
sgn(Rkk+R.) +R.) + + 2sgn 2sgn sgn(R j
j
k,j== 1,2,3; 1,2,3; kk ** j; j; k,j
.
fr r V
\ s * 0. 0
k Ik=i J =i k
U
J
k*j
This Lemma follows immediately from Lemma 11.4.
Lemma 15.2. Let
S(s,t) be defnned
by
(15.13). Then the
ineegral
I H(s,t)|str j S ( s > t ) | s t r 1 1--CGddsdt sdt
(15.16)
cr cr t s
convergss
(15.17) (15.17)
for all
c > 0 if
and only
if
f (
2sgn(R +R ) + sgn(r +r ) * 0, 0,
iI
2sgn(R +R ) + sgn(r +r -1) * 0, 2 3 2 3 3 2 3 2 3
I
II v v
l k 1 k
1 1
k k
f33 ]
3
V sgn(Rk+Rj ) + 2sgn £ k r V £ k r ^- sgn(Rk+Rj ) + 2sgnlk=i
^-
[k=i v v
k,j=l k,j=l
k*j
124
j j
J J
k = 2,3; k 2,3;
ss 0. s0
following
Proof.
Applying
Lemma
11.2 to integral
(15.16),
we have
that
it
converges if and only if
( 2sgn(R +R ) + sgn(r +r -e) < 0, 3 k ° 3 k 2sgn(R +R ) + sgn(r +r -l-2e) < 0, 1 I 1 22 *1 22
(15.18)
kk == 1,2; 1,2;
I 2maxJ 2 m a x | sgn(R sgn(R +R +R ) ) + + sgn(R sgn(R +R +R ), ) , sgn(R sgn(R +R +R ) ) + + sgn(R sgn(R +R +R ), ), 2 2 2 3 \ sgn(R +R ) + sgn(R +R )I + sgn(r +r +r -2c) < 0 ^ ! 3 3 1 1 1 1 2 2 J1 1 1 2 2 3 3 for all all e > 0. From three first inequalities of (15.18) (15.18) it follows that max{ R +R ,R +R ,R +R } ^ 0. max{
Hence
it
is
easily
verified
that
if
A u 2 l2h3 u 3 l3st / 1 inequality • ,-4. of , , • equivalent - , . 4 . the U i C ,o, is e-> +0,t2en (15.18) to last one e +0,then the last inequality of (15.18) equivalentttto (15.17). the last Tois one e-^ (15.17). Consequently, (i5.1o) i1 also is squivalent e(15.17). i1 also oompletes theConsequently, proof of lemma(i5.1o) 15.2. ■ oompletes the proof of lemma 15.2. Note
that
(15.17) (15.17)
follows
squivalent
tt
(15.17).
Tois
■
immediately
from
(15.15). (15.15).
Further
we
need the the following result in our our further studies.
Lemma
15.3. 15.3. Let
S(s,t) S(s,t) be
defined defined
by
(15.13) (15.13) and
R ,r , r ,, kk == 223 223 k
satisfy
the
following
(15.19) Then integral
k
inequauaty inequality 2sgn(R +R ) + sgn(r +r ) ^ 0 . 2 3 2 3
inequality
(15.14) follows
from
the
convergence
of
the
double
(15.16).
Indeed,
it is sufficient
then conditions
to note that
if inequality
(15.19) holds,
(15.17) are equivalent to (15.15).
Here for the space OH" OH"1 (L) we give one main property which
is used
many times in our further studies.
Theorem 15.3. 15.3.
Let Let
1
f(x) G LL )) ,, g(x) G OH'1 (( g(x) € € Jn" Jn"1 ((D L ). . Then Then c c ,? c1 ,? ,? 1 2 2 1 1
function h ((xx)) = — x x h
( 1 D . £\J )
(15.20)
belongs
to
the
1/2
f(x)g(x) f(x)g(x)
space 125
the
5H"1
(15.21)
( L ) = m'1
3
i.e., the
pair
c
3
1
(c ,j
^
3
) is
( L ) U 3JT 1
,3T 1
2
defined
as
2
(L) U W"1 (L), (ci+c2)/2,3ri+r2
follows
3
r (ci>ri) (15.22)
(c ,r 3
(
) = i 3
'
,
if
h(x)
c
< c 2 c < c 2 1 1
if
W
(c^min
Moreover,
if
(yi,y2,yi+y2)),
e
5K * (L) f o r c,r m'1 (L), then ffl-1 (L) D W" 1 (L). c ,y c, y c ,2r 2 2 3 3
aii
if
<^ = c 2
f(x)
€
!JK * (L) c ,y
and
Proof. From Consequence 15.1 and relations of inclusions is
not
difficult
According
to
show
to Definition
that
(15.21)
is
15.2 of the space
equivalent
3ft
g(x)
€
(15.12) it
to
(15.22).
(L) we can represent
function h ( x ) (15.20) in the next form 1/2 (15.23)
h(x)
f*(s)g*(t)x
2 (2?ri• ^
s
\isdt,
t s where (rxcr = {(s,t) € C 2 , Re(s) = Jte(t) = — > . s t 2 By replacing of variable x = s + t - — we can write (15.23) as follows
(15.24)
where
h(x)
F(T)X
2rri
dx,
cr = {x € C, fte(x) = — > and x 2
(15.25)
F(x)
27Ti
f (x-t+—)g 2
(t)dt
for x € cr . X
Consequently, according to Definition 15.2, we have h(x) e 5H some pair
(c,^),
if F(x)|x| e
c
. <= L ( — - i o o , — + i o o ) .
tion (15.25) we get the inequality 126
Using
(L) for
c,r representa
eTrc|T||T|9r|F(T)dT|
7Tc|s+t-l/2| ,
_1
l.y,
*
.
«
r n
,
,. ,
| s + t - — | a | f (s)g (t)dsdt|
e
2TT
cr cr
t s
Since f(x) € JJI (L), g(x) e OH ( L ) , then c ,y c ,y 2*2
1 1
wcjsl e
^ * |s I
wc lt|
f (s) € L(cr )
and
e
*
y . |t| g (t) e L(cr ) .
Hence it is evident that the least double integral converges if
(15.26)
sup
'Hc
e x p U c | s + t - - i - | _ c | s | - c |t| I I | s+t--|- | |s| * | t | 2
( s , t ) €(T x(T s t
')]'
It is not difficult to note that the last inequality is equivalent to
sup
y f ^ ~yi exp c|s+t|-c | s | —c |t| | s+t | |s|
( s , t ) €(T xCT s t
^
"r2 |t|
< co.
'
Hence by using Lemma 15.1 we get that (15.26) holds if and only if
2sgn(c-c ) + sgnt^-r ) < 0, k
2sgn(-c -c ) + sgn(-y -y
(15.27)
k = 1,2,
k
)
< 0,
sgn(c-c ) + sgn(c-c ) + sgn(-c -c ) + 2sgn(y—If
Since
the pairs
(c ,y ) , (c ,y ) are satisfied 1 1
2
-
y ) - 0.
(15.7),
then
is
2
easily verified that conditions (15.27) hold valid for c = c , y = y , where the1 pair (c ,y ) is defined by (15.22). Therefore, we obtain that h ( x ) € m' (L). c ,y 3 3 Now suppose
h(x) = x
1/2
that there is some pair
(c,y) such that the function
1
f(x)g(x) belongs to UJf (L) for all f(x) e 3JT1 c, y
m
(L). Then
c »y
we choose f(x), g(x) such that
V*2 127
1 1
(L), g(x) €
|s |
-TTC
f*(s)
= e
*
-y - l - e |s|
-7ic | t |
1
and
2
g (t) = e
-y - l - e |t|
2
In t h i s c a s e h ( x ) ( 1 5 . 2 0 ) b e l o n g s t o JJJf1 (L) f o r a l l e > 0 . Hence from Definition
1 5 . 2 and ( 1 5 . 2 4 ) - ( 1 5 . 2 5 )
we o b t a i n h ( T ) = F ( x ) f o r x e
Consequently,
+00
>
e n c | t | | i i y i uh* (rx )^d xI
— 2TT
X
1 2?r
exp[ir[c|s+t--i-|-ci|s|-c2|t|)] t s y
-y
-y
-i-e
x|s+t-— | |s| *|t|2 |st|
|dsdt|
for all e > 0.
Since here 2sgn(c ) + sgn(? ) > 0, then 2sgn(-c -c ) + sgnt-jr -y ) * 0, applying Lemma 15.3 we get that the last double integral converges if and only if inequality (15.26) holds. Hence we again obtain conditions (15.27). We shall prove that in this case (15.28)
JJTf1 (L) D UK"1
c*
(L) U Ulf1
c ^
(L) U 3JT1
c2,r2
(L).
( V c 2 )/ 2 , V y 2
Indeed, applying Theorem 15.1 to the two first inequalities of (15.27) we obtain (15.29)
3JT1 (L) 3 3JT1
c,y
c ,r
(L) U JJf 1
c ,y 2
1 1
(L). 2
and, moreover, c ^ min(c , c ). Hence, for the case c < min(c ,c ) from 1 2
1 2
relations of inclusions (15.12) it follows that 3JT1 (L) z> m ' 1 (L). car ( V c 2 )/ 2 ,r i + y 2 Let now c = min(c , c ). Then note that min(c , c ) < (c +c )/2, if
(15.30)
1 2
1 2
1 2
c„* c . Hence in this case inclusion (15.28) follows from (15.29). We 1 2 must prove that in the case c = min(c , c ) = c = c inclusion (15.28) 1 2
is also valid. 128
1
2
Indeed, in this case from the last inequality of (15.27) we get 2sgn(y-y -y ) ^ sgn(c -c) + sgn(c -c) + sgn(c +c ) = sgn(c +c ) < 1, °
1
2
° 1 1
°
2 2
°
&
1 12 2
&
1
2
i.e., i.e., y y < < y y +7 +7 . . Hence Hence using using relations relations (15.12) (15.12) we we obtain obtain that that (15.28) (15.28) holds. holds. Finally, we
see
that
in both cases
inclusion
(15.28) is always
(15.21) it means that JJf1 (L) D 3JTl (L). cy c33,3r ,y3 This completes the proof of Theorem 15.3. ■
valid.
In accordance with
Consequence 15.2. If
both
f(x),g(x) € 3Jt _1* (L), (L), then then xx
1/2 1/2
f(x)g(x) f(x)g(x)
c,y
3JT1 € 3n
c,min(y,2y)
(L).
Remark 15.2. According to Definition 15.2 it is not difficult to note that for A € [R,A * 0, f(x) e 9JT1 (L) if and only if x (A " ~ 1)/2 f(x A ) c,y
JJT* . (L). Consequently, 3Jf* c/ IA|,y | A|,y
a+(j} 1)/2
if we set h(x) = x
A
~
W
f(x )g(x )
€ in
(15.20), where A,w e D OR, Au * 0, then we obtain more general statement than Theorem 15.3. In this case the corresponding pair (c , ,yy )) is is defined defined 3 3
as follows ), [ (c /|A|,y ), l 1
(c 3,y (c ,y3)) == i 3* °3
I
1
if c /|A| < c /|w|,
l
1
(c /|u|,y 2), (c2/|.|,y ), 2
22
2
I (c /|A|,min (y ,y ,y +y )) )) ,, v
§16.
1
22
if c /|w| < c /|A|, ifc/|w|
1122
1
iiff Ci c /|A| /|A| == c 2 //||ww| |. . 2
1
X One-dimensional H-transform i n t h e s p a c e s ffl JJT(L) (L) and 3Jl" 3H a 1( L (L) )
c, y
In §15 (15.3)
and
we have its
introduced
inverse
formula
the one-dimensional (15.5).
In
the
Mellin
present
transform
section
we
establish one important property for this transform and its applications to generalize the H-transform
(15.1) with the Fox's H-function in its
kernel.
129
The results of the present paragraph are very useful for the further studies of the convolution theory considered in Chapter IV. The following theorem is true.
Let in the
Theorem 16.1
strip
s € C, 7) < Reis)
(16.1) a function
F(s) be analytic
Let also the function
function
< 77 ,
and F(s) = 0(|s|~
f(x) be defined
), A > 0, for
by the inverse
|lm(s)|—> oo.
Mel 1 in transform
of
i.e.,
F(s),
(16.2)
f(x) =
l
2ni
F(s)x ds,
Re(s)=v where
77 < v < 77 . 1
Then exists
2
the
and it
Mellin is
transform
equal
(15.3) f (s) of
to F(s) for
all
complex
function numbers
f(x) (16.2) s, belonging
to
77—1
strip
(16.1). Moreover,
f(x)x
e L(0,+oo) for
all
77, such
that
77 < 77
since
the
function
F(s) in
e L(0,+oo) for ri < TJ < TJ . In strip
(16.1)
is
analytic
and
F(s) = 0( I s I " 1 _ ),A > 0, for |lm(s)|—» 00, then equality (16.2) holds for all v,
such that 77 < v < T) . Hence
(16.3)
where x > 0 , 7 7
1
2
|f(x)| < x"y - i ^
|F(s)ds| =
M x
V t
< v < 77 . Further from 77 < 77 < 77 it follows that there
exists sufficiently small c > 0, such that 77 < 77-e < 77+e < 77 . Applying inequality (16.3) for v = 77+e, x > 1 and v = 77-c, 0 < x < 1 we obtain 130
If (x) I < M x _ 7 ) + C ,
f o r 0 < x < 1,
I f ( x ) | < M x" 17 G ,
forx > 1
T7— 1
e L(0,+oo).
Consequently, f(x)x
Now we must prove that if TJ < Reis)
< 7) , then f (s) = 9JHf(x);s} = F(x),
i.e.,
Re(s) +ioo (16.4)
F(s)
X
dX
r
TT
R e (s) + i oo
N
F(T)X
-xdx , =
2ni Re(s)-ioo
lim
x
dx =—7
F(T)X
2ni
N—>+oo
i/N
"dx.
Ke(s)-ioo
The repeated integral in the right part of (16.4) absolutely converges and we can change the order of integration, and after evaluating of the inside integral we have
Re (s) + i oo (16.5)
1 2ni
I(s,N)
Re (s) + i oo
F(T)X
7T1
Re (s) - i oo
Further, using (1937)
and
the
technique
replacing
the
F(x)sh[(s-x)logN]
1
dx
dx.
Re (s) - i oo
of Fourier
variable
in
integrals the
by
E.G.Titchmarch
integral
(16.5)
we
get
the corresponding case of Fourier representation for the function F(s) € L(Ke(s)-ioo, fie(s)+ioo) and moreover it is analytic in the strip which contains the contour of integration. Thus the Mellin transform of f(x) exists and it is equal to F(s) = lim
I(s,N). Theorem 16.1 is proved.
■
Now we consider the classical H-transform and its generalization in the
spaces
JJf^L)
and ffl"1 (L).
In
1970
C.C.Gupta
and
P.K.Mittal
introduced and studied an integral transform whose kernel is the Fox's H-function
defined
by
(1.5).
This
important
defined and represented in the following manner
131
integral
transform
is
(16.6)
(a ,a ) n ' p p H xu p.q (0 ,b ) T T m,
(Hf)(x)
q
f(u)du.
q
Various properties and inversion formulas of the H-function were
also
studied
by
K.C.Gupta
and
P.K.Mittal
(1970), R.G.Buschman and H.M.Srivastava kernel of transform integral
(1.5),
(1970,1971),
transform R.Singh
(1975). Here we note that the
(16.6) is Fox's H-function defined by the contour
which
converges
only
under
some
conditions
of
the
parameters m,n,p,q and (a ,a ), (0 ,b ). These conditions were obtained p
p
q
q
by A.L.Dixon and W.L.Ferrar (1936) and they will been also shown below. Vu Kim Tuan (1986c) generalized the H-transform (16.6) in some space L . In this section we will consider a modification of Tuan's generalization in the spaces 9ft (L) and 3ft (L). First note that if f(u) is replaced by c, y l/uf(l/u) then transform (16.6) can be written in the following form
(16.7)
(a ,a ) } p
m,n p.q
(Hf)(x) =
P
q
where $(s) is defined by
f (u)—
O ,b )
u
=
^-^
2TTI
q
$(s)f (s)x ds,
Re(s)=y
(16.9) below and f (s) is Mellin transform
(15.3) of f(x). Consequently, we introduce Definition 16.1. The H-transform of function f(x),
x > 0, is called
the next integral (16.8)
f(a,a) i,p (Hf)(x) = Hm,n p.q (0,b)
Kf(u)](x)--sr
$(s)f (s)x ds,
l,q
where
= 1/2}
, O^m^q,
m
"TT (16.9)
*(s)
=
j. = i
O ^ n ^ p ,
n
n
r ( 0 +b s) F T T ( l - a -a s ^j
-j
j=i
p
j
j
q
FT r(a +a s) "TT r d - 0 -b s)
j=n+i
j=m+i
132
and the function f (s), generally speaking, is such that f(x) is its inverse Mellin transform (15.5). The parameters of vectors (a,a)
= (a ,a ),
1,
(0,b)
p
p
= (|3 ,b ), . . . , (j3 ,b ), 1
1 , q
are such that a ,...,a ,b ,...,b 1
(a ,a ),
1 1
P
p i
1
q
q
are positive and a ,...,a ,8 ,...,6 q
^
l'
p " l'
'' q
are complex. Moreover, suppose that for them the next conditions take place /te(0.) + -^- > 0, J
j = 1,...,m;
2
1 - Re(a.)-^j
> 0,
2
j = 1,
,n;
(16.10) Re(a
j
) + -i
> 0,
2
1 - /te(|3.)- —
j = n+1,...,p;
> 0,
j = m+l,...q.
Definition 16.2. The ordered pair (K,/I), where m (16.11)
n
£b j=i
(16.12)
p
+ Ea -
E
j=i
P
q
j=i
j=i
q
a - £ b
j=n+i
j=m+i
q
p-q
I a, - E
li = Re
j=i
is called the index of the H-transform (16.8).
The following
theorem gives a relation of the transforms
(16.6),
(16.7) and (16.8).
Theorem 16.2. The H-transform and only
if
the
next
condition
(16.8) exists holds
valid 133
on the
space
3JT (L) if
2sgn(»c) + sgn(jn) ^ 0.
(16.13)
In
this
takes
case
(Hf)(x) e !JJ1 (L). If,
the
next
inequality
2sgn(/c) + sgn(ju-l) > 0,
(16.14)
then
moreover,
place
the
following
representation
of
the
H-transform
(16.8) can
be
obtained
H(*)f(u)^ , u u
(Hf)(x) =
(16.15)
where H(x) is Fox's
Proof.
With
H-function
(1.5).
the aid of the asymptotic
estimation
(4.2) of the
gamma-function and Definition 16.2 for the kernel $(s) defined by (16.9) we get the next relation
| Im(s) | ~M1 ,
<Ms) = o{exp[-nK\lm(s)\]
[16.16)
+co, Re is)
= — , 2
Hence from condition (16.13) it follows that sup|$(s)| < C < +oo. Since f(x) € 3H (L), then from Definition 13.1 we get $(s)f (s) e L(cr), that is why (Hf)(x) exists and belongs toffl(L).
Now if inequality (16.14) holds valid, then from (16.16) it follows that $(s) e L(cr). Hence the Fox's H-function (1.5) exists provided that condition
(16.10) holds. Further
applying
Theorem
16.1 about
Mellin
transform we get that if inequality (16.4) is valid, then
f ,(a ,a ) 1 3H
j H(x); s|
p
p>q
p
(0 ,b ) q
q
134
x
dx = $(s),
for s e cr.
Therefore we can apply Mellin-Parseval equality (15.4) to (16.8) and we obtain 16.2.
representation
(16.15).
This
completes
the proof
of Theorem
■
The following
theorem
gives
an operation
of the H-transform in
m'1 (L). Theorem 16.3. The H-transform (16.12) exists
1
on the space
(16.17)
HTf
(16.8) wihh
the index
(L) if and only
(jc/fi) (16.11)-
if
2sgn(K+c) + sgn(/i+y) > 0.
Furhhermore
theee
is
isomorphism
by H-transform
betwenn
3J11"1 * (L) and c,y
HIT 1
((L). L),
c+K,2C +M C + K , y*\x
Proof. sV
c | ! K
s
The function f * ( s ) « LC<0.
f(x) belongs For f u n c t i o n
JJ1 1 (L) to JTf1
if and only if
c,y
* ( s ) (16.9)
we h a v e
t h e next
eestimation stimation
< 00, oo, s s 6 6 (cr
TT
y
n 7Tc| II m m ss
Hence s e e Hence s L(
° \
I| r* * t,
\*
. t, T
\. •. r>_
J,
i.
-- jj r j* ^r f \ Ml
M 7T(c+K) 7 T ( c + K ) || IInm ll ^^++M s s|| **
f ( (.sj and only o n l y if i f $(s)|s| $(sj|s| f .sj € e Llcrjif Llcr)if and
e e
..
f ((s) sj e e f
On the other hand ||(Hf)(x)|
m *
=
(D J
[|$(s)s^e7r(c+K)|lms|f#(s)ds|
C f[ | Ss ^Ve c7 r| cl |"l ml fs |*f *( (s s) )ddss | t* C 2
and 135
=C C | f ( xx))| | = 22
1 V ) V 1 (L ,D
|s3re7lc|lms|f*(s)ds|
|f(x)||
|$(s)s^e7l(c+K)|lms|f*(s)ds|
E «(Hf)(x)|| 1 UR
(L)
C + K , y+ii
Hence the proof of Theorem 16.3 is completed.
Theorem 16.4. The next
H-transform
-1 (a,a) m,n '1'p|| •[g(u)](x) p>q O.b) l.q'J Hp-n,q-m
(16.18)
q.p
f(|3fb)
1 2711 *(s)
the
inversion
1,m
I (a,a) v
is
,(0,b) m + l,q
for
>[g(u)j(x)
,(a,a)
n+l , p
1,n
g (s)x ds
H-transform
(16.8) in
the
space
!U1
(L). Here
c + K . ^ + fi
(a,a)
and (B,b)
1,P
are defined
and (
. (1 < k ^ j) means the vector
( Proof. belongs *
in accordance
p
k
k> j
In fact, from Theorem
to the space
JH
H-transform
(16.8J
*
and its parameters
of well
regulated
components
z ), . . . , (
j
j
16.3 it follows
(L) and
that
according
g ( x ) = (Hf)(x)
to (16.8)
w e get
-i
g (s) = $(s)f (s). Further [<Ms)] product
with
l,q
have
is also the ratio of gamma-functions the arrangement
which
is defined by
(16.18). Hence the following H-transform (16.18) h a s the index pair (-K,-/I) and (Hg)(x) belongs to !U1 ( L ) . Further substituting the product c, if
136
$(s)f (s) in (16.18) with the aid of the definition of the space 9JT (L) we get the statement of Theorem 16.4.
Theorems H-transforms
16.3 and
16.4 show
■
that
a
is also a new H-transform,
composition
and moreover,
of
any two
the identical
transform (Ef)(x) is included in this class H-transforms.
>["f(u)l(x )
0,0
Theorem 16.5. In accordance
with
= (EfMx) = f(x).
Definitions
16.1 and 16.2 let
f (a',a'! I,P
(H'fMx) = H m , , n , P >q
(16.19)
»|"f(u)j(x)
(0\b') l,q
be some U-transform
with
the
index
(K'./LI'). Then the
composition
of
two
H-transforms
(H'o H)(f)(x) =
(16.20)
exists
on the
following
space
inequality
(16.21)
!JH
c, if
(L) if
is also
H'o (Hf)
and only
if
(x
together
with
(16.17) the
true
2sgnU+K'+c) + sgnCju+fi'+y) ^ 0.
In this
case composition
(16.20) is also
[H>O ( H f ) i ( x ) = H m + m !' n + n ; L J P + P >q + q
(oc,a)
a new
H-transform
, (a' , a' ) , ,
1 ,n
(£,b)
n
, (/3\b')
1 ,m
,, 1,m
(16.22) (ex,a)
, (a' ,a' ) ,
n+l , p m+l , q
and it
maps 3TI
(L) isomorphically
ff(u)l(x)
n +1 , p
, (/3,,b>) ,
0,b)
,
m +1 , q
into
c,J
3J1 c+K + K
137
j
'
, >y
, (L) + ll + l±
Proof. Indeed, here condition (16.17) provides the existence of the H-transform
(16.8) on
isomorphic to UJf1
the
space JJf1 (L). In
(L) by H-transform
JJf1 (L) is
this case
(16.8). Further H'-transform
c + K . ^ + jLl
1 (16.19) (16.19) exists exists on on the the JJf JJff
c+K, "+ fl
is valid. Finally,
(16.22)
(L) if if and and only only if if that that condition condition (16.21) (16.21) (L)
is accounted
by
the
fact
that
the kernel
of
composition (16.20) will be equal to $(s)$'(s), where $(s) and $'(s) arr the kernels of H-transforms 16.5 is proved. The
(16.8) and (16.19), respectively. Theorem
■
following
theorem
gives
composition
structure
for
the
H-transform (16.8). Theorem 16.6. According
to Definionsns
16.1 and 16.2, let
(Hf)(x) k
be
some
H-transforms
k = l,2,...,r. Let kp =PD2,<../'pd hv
C H f H x ) as
dollows
with
also tae
the
the
index
kernel
nrodnck
of
$(s) (16.9) of the*
respectevely
(K ,JLI )
kpmp
the
for
H-transform
1 ) («?) nf
thp the*
(16.8)
H—ti~An
CHfHx) as dollows (16.23) Then,
$(s) = if
the following
r r 1 T T k= =l l' k
conditions
$ k(s). (s). k
are
sasasfied
J J
(16.24)
for
all
Jj
2sgn(c + V K ) + sgn(? + V ua ) > 0 kti k kti k j = 1,2,...,r, then
can be represented
in
the
JJf1 (L) the
space
by the composition
H-transform
of H -transforms
as
(16.8)
follows
k
(16.25) This
(Hff(x) ) (Ho H r r
theorem
immediately
r-1 r-1
follows
0...0Hf)(x). 1 1
from
Theorem
sufficiently remark that the composition (Ho H ^ ^
l
1
16.5.
Here
we
0...<>Hf)(x), 1 ^ 1 ^ r, l-i 1
-
1
1l
exists on the space JJf1 (L)if and only if (16.24) holds valid for all c,7 exists on the space JJf1 (L)if and only if (16.24) holds valid for all j =
1.....1.
j = 1.....1.
138 138
Now we consider four most simple particular cases of the H-transform (16.8), when p+q = 1. Let a,/3 € C, a, b e (R such that
(16.26)
Re(|3) +
— >
1 - Reia)
0,
Then we have four following
[i6 27)
H
o;!(o;b>Hftu)](x)
(16.28)
H
i,o[
-
(16.29,
H
-
=
^7
J ° [ f ^ ) j ( x ) = 2ST
is
S : ? ( ( a ; a ) ) ° [ f ( u ) ] ( x ) = 2si
not
difficult
to
obtain
inversions of the H-transform
(16.31)
H-transforms
H;;;^).^)]^.^
ti6 3o)
It
-
H o.o(, P; »)
1
,1,0 0,1
H°'°f " 1 0,l[(ct,a)J
,0,1 1.0
The following theorem is true.
139
f (s)x
ds
T ( l - a - a s ) f (s)x
I
f (s)x
ds
ds
T(l-a-as)
(13.b)
((oca)}
ds
TO+bs)
that
and
(16.32)
r(|3+bs)f (s)x
(16.27) and
,
- -> 0.
(16.28)
and
(16.30)
(16.29), respectively,
are
the
i.e.,
16.7. Let
Theorem (16.8) with H-transform
the of
index
condition
(16.17) hold.
(K,JU) can
types
(16.27) and
be represented
Then
the
H-transform
by compos it ion
(16.29) as
of
p+q
follows
((a,a)
; b ) H.[ f
p>q
HUO,1((0m+1 ,b m+1 )}
R0,lf(/3
H°>°(
H°>°( 1,0 v (a ,a )
i,o[
(16.33)
_
l,ol(a v
J
,a
n+l
p
n+1
„0,l((a ,a )} oH
i,o[
*_
r
J °---°
u 0,lf(a
,a ))
H
_n
n
i,o(
q
y
>|"f(u)ji (0 ,b )•■■■• H i ; ! o ,b )J-If(u)|(x).
<01 v
1
1
y
v
Proof. Set $ (s),...,$ 1
f 1
,b )
m
m
y
(s) as follows p+q
$ (s)
= r(0 +b s ) ,
j
j
j
j = 1,...,m;
$
(s) = T(l-a -a s) ,
j = l,...,n;
$
(s) =
j = n+l,
m+j
j
j
(16.34) m+j
j
$
j
j = m+1,...,q.
.(s)
P+J
r(l-|3 -b s) j
j
Hence the kernel $(s) of the H-transform product
p;
T(a +a s)
(16.8) is equal
to the
(16.23) of $ (s), k = l,...,p+q. In accordance with (16.11) the
indices K
k
of the kernels $ (s), defined byJ (16.34), can be k
follows
140
written
as
K
= b /2,
j = 1,...,m;
=a/2,
K m+j
= -a /2,
K m+j
j = n+1
p;
J
j
P +
j
n;
J
= -b /2,
K
'*"
j = l
j
j = m+1, . . . , q. J
j
»
» H
Further, it is evident that the index (K,JLI) of the H-transform (16.8) can be as follows (K,M)
Since all a , b j
=
are real positive numbers, then we have K j
> 0 for
k
k = l,...,m+n and K < 0 for k = m+n+1, . . . ,p+q.
Hence, we can write K in the
next form m+n
Ei^i-T E hi-
l*r j=l
j=l
"
j=m+n+l
Therefore, it follows from (16.17) that 2sgn(c+K) + sgn(r+^) m+ n
= 2sgn
-+EiKji j=l
Now
the last
p+q
T
- E i K ji
"
j=m+n+l
inequality
allows us to conclude
(16.24) hold for all j = l,...,p+q composition proved.
We
representation
0.
+ sgn
(16.33)
that
conditions
(here r = p+q). It means that the is valid.
Theorem
16.7 is thus
■
note
that
from
(16.26)
by
Slater's
residue
theorem
(see
O.I.Marichev (1983)) we get
(16.35)
Vl[X
(0,b)J
" Zni
<«•»> W l - s i
>M-
r(/3+bs)x' s ds = b ^ ' " e x p
_,.
.
T(l-a-as)x
141
-s,
ds = a
-1
(a-l)/a
x
exp
[,-<■]
Consequently, Theorem
16.2 (see also
(16.15)) allows us to represent
(16.27) and (16.29) as follows
<"■*" HJ:;U; b ,HH ...».
(x)
,/3/b
exp
-rK)
H^-) ] f ( <
■£;(<■■•>).[«.,]«„ - f.-(i)'""-„[.(i)-'-]„ 0) s
In the case b = 1, /3 = 0 and a = l,a = 1 we obtain modified Laplace transforms which are considered in detail in § 1 7 — § 1 8 .
§17. The G-transform and its special cases
The present section is devoted to the particular case of the H-transform (16.8) when all coefficients a and b are equal to unity. j
j
Naturally, this transform is called G-transform and in accordance with Definitions 16.1 and 16.2 we introduce the following definitions.
Definition 17.1. The G-transform of function f(x), x > 0, is called the next integral
f (a) (17.1)
(Gf)(x) = G m , n p>q
Tf(u)l(x)
where cr = {s e C, Re(s) = 1/2 >, 0 ^ m < p ,
0
77 ro.+s) 77 ra-a.-s) (17.2)
*(s) =
j=i
j=i
p
77 j=n+i
q
r(a.+s) 7 7 rd-p.-s) J
*(s)f (s)x ds, 2712
(0)1,q
j=m+i
142
J
and f (s) such that f(x) is its inverse Mellin transform (15.5). Here we also assume that — -Reioc
/te(/3 ) + — > 0, j = l,...,m; ' j
(17.3)
,
2
Re{ocJ+— J
J
2
) > 0, j = l,...,n; j
> 0, j = n+l,...,p; —-fie(/3j > 0, j = m+l,...q.
2
2
J
Definition 17.2. The ordered pair (c ,^ ), where P
p+q 2 '
(17.4)
q
Re J=l
J=l
is called the index of the G-transform (17.1).
From Theorem 16.3 it follows immediately that
Theorem 17.1. The G-transform on the space Jit
(L) if
and only
(17.1) with
the
index
(c ,f
)
exists
if
C, Q
2sgn(c+c ) + sgn(^+^ ) > 0.
(17.5)
m
In
this
#
#(D.
c+c
case
the
G-transform
maps 5J1
(L) isomorphically
into
,v+y
Now applying Theorem 16.4 we see that the inversion of G-transform (17.1) is also G-transform and it can be written as follows -1 ■
G m,n
p.q
((a)
I,P
V
•(g(u)j(x
(£) 1 ,Jq ^
y J
(17.6)
= Gp-n,q-m
q>p
m+1
(a)
'q
1 > m
, (a)
n+l , p
2ni
hHs)
og( U )
IL
1,n '
g (s)x ds.
143
(X)
J
The following theorem represents our G-transform defined by contour integral (17.1) in the traditional real form.
Theorem 17.2. Let (17.7) then the
the real
2sgn(c ) + sgn(? -1 ) > 0, G-transform form
as
(17.1) with
the
r
(17.8)
index
(c ,y ) can be represented
in
follows
(Gf)(x) =
(a) }
X
G m,n
p
du f(u)
p.q u
where
Gm,n "p.q
(a) 1 p
is
Meijer'
s G-fund
ion
(1.3).
(0)
Here the existence of the G-transform (17.1) is guaranteed by (17.7) and inequality (15.7) in Definition 15.2 of IJJf1 (L) (see also (17.5)). Theorem 17.2 shows that inequality
(17.7) provides the convergence of
the Mellin type integral (1.3) which is used for definition of Meijer's G-function at least for the contour L, coincided with the line
Remark 17.1. If inequality (17.7) is replaced by 4sgn(c ) + 2sgn(3r ) + sgn|p-q| > 0 then the statement in Theorem 17.2 is also true (see Vu Kim Tuan, 0.1. Marichev and S.B.Yakubovich (1986)).
Note that classical G-transform (17.8) in the next equivalent form r
* (17.9)
(Gf)(x)
_m,n G xu p.q
,
was also
studied
Kesarwani
(1962;
by
several
1963
(a) ) P
f(u)du
q'
mathematicians
a,b;
1965
V.K.Kapoor and S.Masood (1968)). 144
a,b;
(see C.Fox
1971),
(1961),
V.K.Kapoor
R.N.
(1968),
Further, since the Meijer's G-function
(1.3) is rather a general
function, then from (17.8) any its particular case defines the corres ponding transform, which may be known or unknown. Here we give the table of those transforms which are required to study the convolution theory in Chapter IV. For convenience we introduce the following notations of transforms
(see also Brychkov Yu.A., Glaeske H.J.
and Marichev 0.1.
(1983))
(17.10)
k( x)joff(u)l
(17.11)
x \ ( x ) U f ( u ) j = r ^ j a k ( 5 ) f (U)«H = x a | k ( x ) U u - a f (u)
k(*)f(u)*H u u
2ni
k (s)f
(s)x ds,
= (xak(x)x"a)(f(x)), - 1
1
(k(x)} .[f(u)] =-±j
(17.12)
*
k (s)
-s
f (s)x ds,
- 1
-1
(17.13) (x\(x)} .[f(u)] = ^ i J
Below
we
give
table 17.1
1
f (s)x ds = (x k (s+a)
of
definitions
of
a
{k(x)lx-a )(f(x)).
important
simplest
G-transforms in forms (17.10)-(17.13), where the function *(s) (k (s)) is defined by
(17.2) such that p+q ^ 2. These transforms are special
modifications of known classical integral transforms and their inver sions
(Laplace
transform,
Hankel
transform,
Stieltjes
transform,
Riemann-Liouville integro-differential operators, Meijer transform). The more general particular cases of G-transform, which will be introduced in Chapter IV, can easily be obtained from (17.1), (17.10)-(17.13) by the using the table of Mellin
transforms and representations of the
kernels
G-function
k(x)
through
Meijer's
(1989)). 145
(see
A.P.Prudnikov
et al.
TABLE 17.1 Simplest integral G-transforms and their inversions
Modified Laplace transforms
00
■ <°[ 2
G°'*
oI
-(x/u)_, .du f(u)—
.ff(u)l(x) = { « " * } • [f(u)] " < A / H x > = f 0 00
1] .[f(u)](x) = l e " 1 / X H f ( u ) ] = (A-f)(x)
=
f
-<x/u)~*f (.du u)—
0 0
3
' -(x/u,*1 -«fl ,du {xVx±1}o[f(u)] = ( x < V a ) ( f ( x » = x * e u flu)— c) -1
4
1 ~" r 1,0 G
o,i
I°
»[f(u)](x)
= { e - * } »[f(u)] = (A;'f)(x)
= _i_
[_JL_ f*(s)x-sds
J r(s)
5
f V x _ 1 l .[f(u)l = (xV^HfCx)) - - J L
1 f*( )x_sds J. 1 Os J Jv U O T(±(s+a))
——^^^———^^^—
I
146
Modified Riemann-Liouville integro-differential operators and their inversions
6
GG l° '*l jf[
a a
X "1)++ l| °o [[ ff (( uu) )]l((xx )) == J| ((X "1)
|orf(u)l == I^(xI^(x-aaf(x)) f(x)) V[f(u)]
x
r (x-u) a_i -«_ ,_,
= [ ' x - " ) 0 6 " 1 u-«f(u)du, = J T(a) u f(u)du, Jo T(a) o x
Re(«) Re{a) > 0
x
1 nn - «f(u)du, , . ,A,A = d f ( xx--uu))""**" - u-«,, = dxnJ Ha+n) u f(u)du, dxnJ r(a+n) oo
Im(oc) 1,2,3,.. 7/n(a) * *0,0,n n= =1,2,3, ..
-n < Reioc) Reia) < <-n+1 -n+1 oror ReRe {a){a) = =-n, -n, 7
Gl jf ':lPL f G!'?[
aa
|o[f(u)l(x) = j
1, [[ 0 J L J
X )) ++ (( 11 X
"
([
lo[f(u)l = Ia(x"af(x))
r(a) f(a) J L J
00
=
l u XJ
u'af(u)du,
r(a) J r(a)
/te(a) 00 Reioc)> >
xX
00 00
ff = =
dd ^[(u-x) ^[(u-x)
a+n_1 a+n_1
l" r ^^ JJ JJ rr(a+n) (a+n) x
- n < Reioc)
< -n+1
- n < Reioc)
< -n+1
or or
-a-, ,. , , -a-, (UMU
" f(u)du' ' ' u
x Reioc)
= - n , I/n(a)
* 0, n = 1 , 2 , 3 , . .
Re(oc)
= - n , I/n(a)
* 0, n = 1 , 2 , 3 , . .
-l
8 8
|
(±1+X X ))
+*
|1 . [ f (l u ) ] = ((l^)"(x-af(x)) l ^ ) " ( x - a f ( x ) ) = i ; a ( x "- a f ( x ) )
147
Modified Hankel transforms and their inversions
9
1,0 0,2
|~f(u)l(x)
G
= j j ^ U v S o j o |~f(u)l
iV2,-iV2
J V (2/^7U)f(u) —u
10
0,1 G, 2,0
1-IV2,1+IV2
M ( x ) = -KbHHH J (2/~TI^Of(u) — V
u
-1 11
a,o J
0,2
Tf(u)
J {2Vx)\
(x)
o f(u)
V
iV2,-i>/2
'-,J.tHM'H -1 12
.0,1 J 2,0
fl-iV2,l+iV2)
■Htx) = Wi)} °M xJ ( 2 \ / x ) U | f (u)l
148
Modified Sin- and Cos- transforms and their inversions
13
G
1,0 0,2
Tf(u)j(x) = j—sin(2v^)|o["f(u)l 1/2,0
_1
14
G
^ 1,0 ( |"f(u)l(x) = |—cos(2V^)|orf(u)l 0,2 0,1/2
_1
15
0,1 G. 2,0
0,1 G.2,0
cos(2i/x/u)f (u
du
f 1/2,1 o|"f(u)l(x) = j—sin(2x'1/2)|o["f(u)l
_1
16
,du sin(2yx/u)f(u)— u
sin(2/u/x)f(u)du
( 1/2,1/2 »["f(u)j(x) = j—cos(2x"1/2)|orf(u)l
J. cos(2^/u7x)f (u)du
149
r
r
17
(/
|G^O G°;J
-1 -i \\ ~\
--1l
Mof [(fu( u) )] ]((xx)) = | - i s i n ( 2 vV^x))}|
=
o[f(u)] .[f(u)]
M / ^ x -x1-/ 12/s2lsni n( 2( 2xx- -l /12/ 2) )l loor[ff ( u ) l U? JL ViF J L J J
-l -1 18 IB
l l o .frff((uu))ll (( x ) -=((^- ci coos s( (22VVxx))\\11..rf ff (( u ) l [V G ° M A ~ 11 L J 1J 1 |[ 22 '' 00 (l oo ,, ii // 22jjJ| L J l[^ /i L 1 J ( 1 i -1/2 -i/2 r,., J -1/2.1 r\el, ,1 o -1/2,1 <— x JM ) ) = ^— cos(2x H ° f (ful u
-1
19
k'jf 1/2,1114f(u)](x) = {-sin(2x"i/24 °[f(u)] [ G ° ' j [ 1 / 2 , 1 | | oTfCujjCx) = U s i n ( 2 x - 1 / 2 ) |
o[f(u)]
/2 = jp- A x sin(2^)}o[f(u)l = x11/2 sin(2^)}.[f(u)l
20
20
-1 r 1 r 1/2,1/2 n
-1 1 f n ,( 1/2,1/2 l l r i /• , i"1 r i 1/2 n r G°;J . [ f ( u ) ] ( x ) = | - i c o s ( 2 x -1 / 2 ) | . [ f ( u ) ] G°;J o [ f ( u ) j ( x ) = j - i c o s ( 2 x - ) | o[f(u)j
n
/-
r r i
= _ ! x 1 / 2 cos(2yx)}o[f(u)l cos(2v^)|orf(u)l = j|_L
150
Modified Stieltjes transform and its inversion
21
G
1,1 { 1-P 1 >|~f(u)l(x) = jr(p)(i+x)~pjorf(u)l 0 1,1
uPf(u) du
Tip)
(x+ur
,1,1 '1,1
22
1-p 0
>[f(u)l(x) = | r ( p ) ( l + x ) " P |
o|"f(U)l
1
1
2ni
r(s)r(p-s)
f (s)x ds
Modified Meijer transforms and their inversions
23
G
2,0 0,2
>|~f(u)~|(x) = J2K y (2v^)|orf( u )l y/2,-i>/2
= 2
24
0,2 G. 2,0
l-iV2,l+y/2l
K (2/^7U)f(u) — V u
M(x)-Kkr)}= 2
f(u)
Kv (2\Hi7x)f(u) du
151
25
.2,0 J 0,2
>|"f(u)](x) = J2K^(2V^)|
1 2ni
26
.0,2 J 2,0
o|"f(u)l
v/2,-v/2
f l - v / 2 , l+v/2'
r(s+iV2)r(s-i>/2)
f (s)x
ds
■H(x)-htr)} °M i 27T1
1
r(-s+v/2)r(-s-v/2)
Integral transforms with Kummer
f (s)x
F (z) and 1 1
Whittaker
27
W
(z)
functions
G 1, 1 [ 0 ^ _ b J o [ f ( u , ] ( x , = { r [ ^ ] i F i ( a ; b ; - x , } « [ r ( u ) ] 1,2 00
= r [ M [ F (a;b;- 5) f(u )du [_ b J i i
28
1,2 G. 2,1
u
u
fl/2+v,l/2-i/l ff(u)l(x)
= jr[l/2-p-y,l/2-p+Je(2x)
-p 00
W p ^(i)Wf( u )j = rJl/2-p^,l/2-pJ ie (2x/u)
152
.,
W p,v
,U,._,
.du
(-)f(u)— x u
ds
§18. Composition structure of the H- and G-transforms
As it was introduced
in §17, the modified Laplace transforms with
power multipliers and their inversions are defined as follows
[18.1)
(xaA x"a)(f(x)) = +
^
f(s+a)f (s)x ds
2ni
e'x/u u~af ( u ) ^ , u
(18.2)
ReU)
> -1/2;
(x"aA xa)(f(x)) = - U r(a-s)f (s)x~ ds Zni
-u/x a_, sdu e uf(u) — , u
(18.3)
, a -l -a., _r . . 1 1 f (s)x (X A X ) (f (X) ) = ^—r + 2ni f(s+a)
(18.4)
(X
A
X ) (f(X) ) = ^ r
2ni
They
are
simplest
particular
, . Reioc) >
ds, ReU)
/0
1/2;
> -1/2;
1 f (s)x~ ds, Re(a) > 1/2. f(a-s)
cases
p+q = 1.
153
of
the G-transform
(17.1),
when
According
Definition
17.2,
the
corresponding
(18. 1 )-(18.4) are equal to (1/2, -Reoc) ,
transforms Reu)
to
indices
(1/2, 1-Rea),
of
(-1/2,
and (-1/2,-1+Rea), respectively. Hence they map the space 3J1
(L)
c , if
i s o m o r p h i c a l l y i n t o IJJf1
( L ) , JJT 1
c +l/2,3T-Rea
JH
fflfl"1
(L),
c +1/2, y + l - R e a
(L),
c-l/2,2r+Re(X
( L ) , respectively. c-i/2,r-i+Rea In the present
G-transform
section we find composition
representations
by these simple transforms. Here we also give
of the
composition
structure for all known convolution type transforms which are mentioned in §17. In accordance with (18.1)-(18.4) we index all operators of the form
x
13. - £ . J A x J, j = 1,
i-a
J
. . . , p and x
J
x A
x
J
a -l a_ -a A x J , j = l , . . . , n ; x J A _ 1 x J , j = n+1,
,j = m+1,. .. ,q in a single sequence A ,..., A 1
According
to
Definition
17.2 denote
by
# *
(c ,3- ) k
transform A
for k = 1,
p+q
the
index
of
k
p+q respectively. After some calculations, we
k
get f 1
(c ,7 (c
j
A
,y
m+j
(18.5)
(c
m+j
(c* p+j
From
) = (l/2,-Re(/3 )]
j
fy
for j = 1,...,m;
j
m+j
m+j
,/
) = (l/2,Re(a ))
for j = 1,...,n;
) = (-l/2,/te(a ))
for j = n+1,. . . ,p;
) = (-l/2,-/te(|3 ))
for j = m+l,...,q.
j
j
p+j
j
Theorem 16.6 about
composition
representations
of
the H-
transform it is not difficult to obtain the following result.
Theorem space
1
3JT
permutation
18.1. Suppose
that
(L), i.e., conditions
{j , . . . , j 1
} of
the
G-transform
(17.3) and
(17.5) hold.
the sequence
p+q
conditions 154
(17.1) exists Then
in for
{l,...,p+q> satisfying
the any
the
(18.6)
2sgn c + V c.
the following the
space
+ sgn Ur + V" y.
composition
representation
£ 0
for r = 1, . . . ,p+q,
of the G-transform
is valid
in
!ffi (L): c,y
(18.7)
Remark
„m,n J p,q
18.1.
(
(a) 1,P
o|"f(u)l(x) = [A O ...O A f)(x) L J I Jp+q \ J
l,q
Here from
(18.5) it follows
that
c
= 1/2
> 0 for
j
j = l,...,m+n. Hence composition representation (18.7) is valid for the permutation {j ,. j > = {l,...,p+q> (see also (16.33) in Theorem p+q
16.7).
Remark
18.2.
By
combining
the
operators
A
.,A
,
with
preservation of their order, into various groups, we can obtain other composition representations of the G-transform, having, in particular, Riemann-Liouville
fractional
integro-differential
operators
and
the
Hankel, Stieltjes, Meijer transforms, etc.
Example 18.1. Now we consider some composition representations of G-transform with Gauss hypergeometric function (1.1) in kernel
(18.8)
F (a,b;c;-x]
2 l
Tf(u)
F (a,b;c;- - ) f ( u ) —
2 1
155
u
u
According to Table 17.1 of the definitions of simplest operators and the following Mel1 in correspondence between the kernel
F (a,b;c;-x) and its
Mellin transform (15.3) (see formula 8.4.49.13 in A.P.Prudnikov et al. (1989))
IT ( u ^ = F (a,b;c;-x)
2 1
Ml-a,l-bj
T(c)
G 1,2(
r(a)T(b)
Z
'
Zl
(18.9) s, a-s, b-s
r(c) r(a)r(b)
we have the next composition representations
j2Fi(a,b;c;-x)|orf(u)l
r (C)
, . , , -a. a. , (A )o(x A X )o( x A x
)o(x°A
x°)of(u)
r(a)r(b)
= usi (x 1 - 0 ^- x - i )./(i + x)- b i.rf(u)i
(18.10)
T(a)
r(c)
+
I
jx(1-c)/2J
rta)r(b) I
/ L
J
(2^)U{x-(a+b,/2K
c a
"
/ I
(2x-1/2)lo[f(u)].
a b
"
/ L
Below we give the table of composition representations
J
(factoriza
tions) of simplest operators by modified Laplace operators. Here symbol f(u)
in the left and the right parts of equalities given below is
omitted.
156
TABLE 18.1 Factorizations of Simplest operators
1
/ x V x ) W x PP h ( x ) W f ( u ) ]
2^
a a -a-a x Vi xx"a _ a x i ++ x
= x aa | k ( x ) J o / x // 33 - aa h ( x ) U u - // 33 f ( u ) ]
_ a+a-i 1 i-a-a a-i - i1 1i - a , - ixf x a + a "A_X A x 1 " a ";ao)^ox( x a "AV X x " a ); ix A_X ;o^x A X ;
3
x - 4 - = (x _ 1 A x ) o ( A _ 1 )
3
x - 4 - = (x _ 1 A x ) o ( A _ 1 )
{j^(2^)l = ( x ^ A y ^ ^ o C x ^ - ^ V ^ 1 ^ 2 )
4
5
fT , „ - 1 / 2 . ) |Jv(2x )| f,
,„
-1/2,1
, -V/2A V/2, , 1+1//2 -1 = (x Ax )o(x A+ x , -IV2.
V/2,
, 1+IV2.-1
-\-V/2. ) -1-IV2,
-l
5 6
| -j M = (x j y2(x2 ^ ) | ) | = ( x - 1 - yA/ 2xA _ x )1 .^( /x2 ) o ( x i ;A/ 2+ Ax- 1 x - y / 2 )) -1
6
{V2^1}
7 7
-l fT , „0 - 1 / 22 ,. 1) f, |V2x >J |j^(2x )|
8
99
= (x~1~lV2A/+lV2>°(xl'/Vx~V/2) y //22A. / /22 . , --IV2 V/2. , l+V - 1 - I1V I V 2 . --11 I V 2. = (x \Ax )o(x A_ >) A xX +x
{( -Ji L- s i n ( 2 vV/ x ) l = ( x 1 / 2 A xx"-11// 22 ))o. ((xx-"11AA" ^1 x )
|/ _ J! _ s i n ( 2 x -- 11 // 22 )) l| = ( x "- 1 / 22 A __ xx11//22))oo((xx A A^' xV"11 )
- 1l
10 10
/| A J _- s i nn((22V^x)) ll \.rz
2 A x )Joo( xt x1 / ^ AV"11 = ( x ^- V *
157
x"1/2)
11
1
—
,0
-1/2,
,
) \
c o s ( 2 x - 1 / 2 ) l = (A
{-i-
.
-1,
= (x A x
, -1/2.-1
)o(x
A
1/2,
x
c o s ( 2 V ^ ) l = (A ) o ( x " 1 / 2 A " 1 x 1 /
/J_
12
13
.
sin(2x
)o(x1/2A-V1
-l
—
14
15
16
17
18
19
cos(2\/x)i
1
,_
cos(2x
f(p)
Tip)
, -1/2.
= (x
-1/2. 1
)}
1/2,
Ax
, 1/2.
= (x
,A"U
)o (A
-1/2,
Ax
)
, -1,
)<>(A_
( l + x ) ~ p i = (A + )o(x" P A_x p
1
(l+x)~Pt
= (A"1)o(x"pA"1xP^
r V/2. «' / *A - 1 V/2, --V/ = ( x Ax )o(x A x
J2 ^ ( 2 x " 1 / 2 ) | = (xy/2A_x" V / 2 ,
20
Kv(2i/x)|
21
2 V2x- 1/2 )j
, "1^/2 A
) o (x
V/2,
Ax
)
, V/2 -1 - V / 2 , , - V / 2 -1 V / 2 , (X A X )o(x A X )
, -V/2 -1 V / 2 ,
(x
158
A x
, V/2 -1
J o (x
A x
-V/2,
)
The next table contains Mellin transforms of the kernels for simplest
operators,
which
give
the composition
the
representations of
G-transforms. All correspondences can be obtained by the Mellin-Parseval equality (see also (17.10))
k*(s)f*(s)x s ds.
(18.11)
L
TABLE 18.2 Mellin transforms of the k e r n e l s for the Simplest
1
kk**((ss)) < <
2
k*(s+a) <
. 11 m
3
kk**((ss++aa))
A 4 4
r1 (<*+<*) is+aj 1 is+aj
operators
>> J| kk (( xx ))l|
> | x aa k ( x ) |
< < >> jxjx0C0Ck(x)}" k(x)}"11 I I
4 < <
J J
»>/*a * x e"x e 1 |- xaA x A + x"a X >| x e | - x AA+x+ X j
> |{xx--aaee--11// XX lj = x"aA_xa
5 5
r(a-s) <
6 6
r(s+a) r ( s + a ) *< Hs+a)
x x *> i1 xX ee / } "" x \♦x
1 , « r(a-s) * * ru-s)
, / - a -i/x\" -i/x\_1X _ - a - li a > \ / ~ " x A" x * \x e /
7
1
{a a -x\
159
_ a -l -a
T(a+s)
Jxa(l-x)^~a M _
r(a-s)
Jx^^x-l)^-1!
a |3-a -(3 -0
8
9
^-
T "~ ^
r(a+s) T(a+s) 10
rnoe - s ) *<
10
11
12 12
113 3
.14. 14
1C 15
/ (a-/S+i)/2. (a-/3+i)/2T
}
i - ^T P-a i-P - a a-i
I++ xX
r, o, / - ^, !l
J
>\* > \x
= x =X
UVx) a*P-ilZVx) f oc+p-i
■»M,iw4**lw ^ 2 ^ 1 — {^"~-^4 r ( 3S / 2 - «g - s ) <<
T(a-s) r(a-s)
r(a-s)
ro/2-a+s) '<
16 16
r(a+s)r(p-s) r(a+s)r(|3-s) <
r(a+s)ro-s)
L -1/2II -l/al)
J
a.p-i[2x
*{*
r(a-s) r ( l / 2 - a + s ) *<
1^
» {{-— ^ -xxaa'-11//22ssiinn((22 v ^ ) l|
>
/ 3-a-i>/2, 0-a-i)/2 T
IT^iT'
15
17 17
|
r (P-a; M n
))
/ 1 -a ,„ --1/2,1 1/2,1 * \ ^ X C°S(2X >J7
/ 1 -a+1/2 . r._ -1/2,1 o -1/2,1
*> \^rx \~x
nt s iin( 2x
'J '[
a a p > |r(a | r ( a ++ p)x | 3 ) x a(i+x)" ( l + x ) " a ""/ 3 |
1 <—* ^ _ / r ({r( « a+,„x*u x « ( i + xr«-nx)-n" v
160
+P
+
1
J
(a+/3)/2
r(a+s)r(/3+s) << r(a+s)r((3+s)
> | 2J2x » x(a^)/2K
r(a-s)rO-s) <
> |2x-(a+/3)/2Ka_/3(2^)i
19
r(a-s)rO-s) <
» |2x- ( a + P )/2Ka/3(2^)|
20
1 r(a+s)rO+s)
18 lo
19
I
1
20 21 2i
z <
,
r(a+s)r(0+s) < r(a-s)f(p-s) < na-s)r(e-s) *
22
r
23 23
r(s- P )r(i/2-i;-s)r(i/2+y-s) )r(i/2-v-s)r(i/2+v-s) <-
K aAZtfi)] / 3 (2^)|
(X-p
J
, / (a+0)/2 r^/zA * \2 X Ka-|3 t2Vx 7
>
„ / , (a+£)/2
OJZA'1
*\ 2 X a-pl2l/x)/ 2^ x ^( ^ a +^P V , / / S (^ > f »{ " \ - p 2 ^^ )^}
nbrSs) — ( ^ i . V ^ ^ 1 }
161
CHAPTER IV. GENERAL INTEGRAL CONVOLUTION FOR THE H-TRANSFORMS
As is widely known, famous Laplace, Fourier and Mellin convolutions, which also appeared in the last century, have important applications in various fields of mathematics and physics (for example, E.C.Titchmarch (1937),
I.I.Hirchman
and D.V.Widder (1955),
I.N.Sneddon (1951; 1972).,
N.Wiener (1937), etc.). Except these classical convolutions, there are convolutions
for other
integral
transforms
and
series
in
literature
(N.Ya.Vilenkin (1958), V.A.Kakichev (1967; 1990), M.Flensted-Jensen and T.Koornwinder (1973), I.H.Dimovski (1982), H.-J.Glaeske and A.Hep (1986; 1987; 1988), S.B.Yakubovich (1987), etc.).
Recently S.B.Yakubovich
(1990a) introduced new general convolution
(f*g)(x) which is defined by double Mellin-Barnes contour integrals,who se theory being considered in detail in Chapters I and convolution contains a great number of particular cases
II. This general for Mellin type
transforms, the classical Laplace convolution is one of them.
The
basic
factorization
property
for
this
convolution
is
the
following
(K (f*g))(x)=(K f)(x)(K g)(x), 1
where K , K , K
2
3
are some one-dimensional transforms which, in general,
may be different.
In the present chapter we will consider this general convolution in the spaces 3JT (L) and JJ1"1 (L), (introduced in §15) for the case when c, y K , K , K are some H-transforms which are considered in §16. Various examples
of
convolutions
for
the
obtained.
162
classical
integral
transforms
are
§19. Classical Laplace convolution and its new properties
As is known, the Laplace convolution of two functions f and g is defined as follows
(19.1)
For
the
(f*g)(x) =
above
convolution
f(x-u)g(u)du.
there
exists
the
following
factorization
property
(19.2)
(L(f*g))(x) = (Lf)(x)(Lg)(x),
where L denotes
the classical
Laplace
transform
(see V.A.Ditkin and
A.P.Prudnikov (1965))
(Lf)(x)
(19.3)
e
f(u)du.
Now we suppose that f(x), g(x) belong to space !Jft
(L) defined in
c, 7
§15. Then representing f(x) and g(x) by the inverse Mellin transform (15.5) of f# (s) and g* (t), respectively, and after changing of the integration order and using
the beta-integral
(see A.Erdelyi
et al.
(1953) V.l.) we can represent (19.1) in its equivalent form as follows
(19.4)
(f*g)(x) = (2TTI)
2
rd-s)r(l-t) * , *,., -s-t, .. ——r= ^ — f (s)g (t)x dsdt, f(2-s-t) t s
where it is to be remembered that,
t
= 1/2}. For Laplace convolution (19.1) we have the following result. 163
= ^e(t)
L
19.1. The
Theorem exists
for
all
classical
f(x),g(x) € 'JTI
Laplace (L) and
convolution
it
possesses
(f*g)(x) the
(19.1)
factorization
c , "%
property
(19.2). L
— 1/2
In
this
case
-1
(f*g)(x) e !ffi
x
&
if
x~
(f*g)(x)
€ HIT1 ( L ) ,
e JR~ V
y
(L).
Furthermore,
c,min(y+1/2,2^+1/2)
(L) f o r some p a i r i
(c , ^r ) and f o r a i i * *
f(x),g(x)
then jjf 1 c
Proof. From
( D D UJT 1 , 3T
(19.4)
(L).
c,mi n ( y + 1 / 2 , 2 ^ + 1 / 2 )
it
follows
that
the
Laplace
convolution
L
(f*g)(x) exists if and only if
(<*n ^ (19.5)
r(i-s)rd-t) * . * , p ^ —. f (s)g ( t ) € I (.Z-s-t)
Tr X(T ), L((T s t
Using the asymptotic estimation for the gamma-function we obtain (see, also (4.2), (4.3) and (4.10))
(19
'6)
n
r(2-s-t)U "
H
l .. I -1/2 S+ t
>[-5('
exp|-5| |s| + |t|-|s+t,
where E = const, s, t — > oo, (s, t) e {
M O 7^ (19.7)
r(i-s)ru-t) sup (s,t)€((T x cr ) s t
r(2-s-t)
y <
+W
-
On the other hand f(x), g(x) € JJf1 (L) then
(19.8)
expire] |s| + |t| 11 |st|y f*(s)g*(t) e
164
Lir^xcr^
Since 2sgn(c) + sgn(y) ^ 0 (see Definition 15.2),
*
then it follows
*
(s)g (t) (t) e<= L(cr L(cr x x cr cr ), ) , which, which, together together with \ that f f (s)g s
t
(19.7),gives (19.5
L L
It means that the convolution (f*g)(x) exists.
Now by the replacement
x = s+t-1/2
representation
(19.4) can be
written in the next form L
(f*g)(x) (19.9) 2ni
r(3/2-x)
1 ni/2-T+t)ni-t)f (x-t+i/2)g (t)dt, 27ri
dr
where cr = {x € C,Re(x) = 1/2}. According to Definition 15.2 and from L
representation
(19.9)
it
follows
that
x~ 1/2 (f*g) (x) e JJT1 c
(L)
if
and
i ,yi
only if
F(x) r(3/2-x)
(19.10)
e
*VT' *i w , iTl € L ( ( r x } '
where F(x) denotes the inside integral in (19.9)
(19.11)
It
is
F(x) =
evident,
that
27T2
this
r(l/2-x+t)r(l-t)f (x-t+l/2)g (t)dt.
integral
converges for
all
t e cr .
Further,
from (19.10) we have
.
2ni
.
F(x) e r(3/2-x)
_1 271
rn
He T
l
r
, | l x dx
n ^ r M f^>
r(i-s)r(i-t) —^j2 p j —e cr cr t s
71c s + t - 1 / 2
i
y
i . . ._ i i_* . * , , , , ,, s+t-1/2 f (s)g (t)dsdt
165
From (19.6) it follows that the last integral converges if and only if
K -1/2
exp
[«h
>)]
+l/2)|s+t|-l/2|s|-l/2|t||||s+t| *
f*(s)g*(t)
(19.12) <= L(cr x cr ) s
t
Since f ( x ) , g(x) e 311"
(L) then from
(19.8) we have
that
the
relation
c, If
(19.12) holds valid if
{
sup
-i/«i
5
|s+t|
r
|s|_3r|t|"^
*
/
e x p 7r ( C i + l / 2 ) | s + t |
(19.13) -(1/2+c)
s -(1/2+c)
t
I < +00.
By applying Lemma 15.1 to (19.13) we get that the last inequality is equivalent to 2sgn(c -c) + sgn(? -y-1/2 ) * 0, 2sgn(-l-2c) + sgn(-2^) < 0, 2sgn(c -c) + sgn(-l-2c) + 2sgn(* -2^-1/2 ) < 0.
Since
2sgn(c)+
sgn(^)
>
0,
then
the
last
set
of
conditions
can
abbreviated as follows
2sgn(c -c) + sgn(* -y-1/2 ) ^ 0, (19.14) [
It
is
s g n C c ^ c ) + 2sgn(r -2y-l/2 ) < 1/2.
evident
that
conditions
(19.14)
hold
valid
for
c
= c, l
y
= min(y+l/2, 2y+l/2). Consequently,
x" 1/2 (f*g)(x) € 3JT1
#
c,min(y+1/2,2fr+l/2)
166
(L).
be
L
Now let x-1/2(f*g)(x) € ffl"1 (L) for some pair (c ,y ci
), such that
1I
2 s g n ( c ))++ s g n ( * ) > 0. We must show t h a t
m'1
(19.15)
< =11»y » y ii <=
In fact, since defined
by
(L) (L) D m'1
ffl"1
c , m i n (( yy + 1 // 22 , , 223yT++l1//22 ) c,min +1
(L). (L).
C | s l" l 1~ 1_e L(cr )) for for all all ee >>0, 0, then then the the function function h(x), h(x), I e e L(cr
the
inverse
Mellin
transform
from
h*(s) =
x~1/2(h*h)(x) Consequently, we get x"
e-irc|s| ,,-y-i-e» belongs to jjj-i (L) 1 1
(15.5)
c>7
Gm'1
(L). But, as is shown above, this is equivalent to
(19.12),
1 1
* where
* and
* following
definite
form
f (s) =
where f (s) and g (t) have the following |-y-)-a gg.*((tt)) == ee-nc|t| -iic|t|| e-iic|.|| t|-e-i-e -*d.| |s|s-,-)-«a_ -e-i-C> .. ee e |t|
definite
form
f (s) =
[
f (s)
g (t)
have
the
,
71 (Ci+1/2)|s+t|-(c+l/2)|s|-(c+l/2)|t|
€ L(tr L(cr xcr )) s
N-J
|s+t|
7If 1-1/2 -1/2
|st|"^|st|_1" |st|~*|st| ~G
for all c > 0.
t
Since here 2sgn(-l-2c) + sgn(-2*) < 0, then we can apply Lemma 15.3. Hence we obtain inequality (19.13), which is equivalent to (19.14). From the first condition of (19.14) it follows that c
(19.16)
If c
l
ss c. If c
< c, then
2sgn(c-ci) + sgnmin(y+l/2,2y+l/2)-yl ^ 0.
= c, then (19.16) follows immediately also from (19.14). Now from
Theorem
15.1 and
inequality
(19.16) we get
(19.15).
Now we give new
simple proof of the factorization property (19.2). First note that for f(x) € € 5n-1(L) the classical
Laplace
Parseval equality in the next form 167
transform
can be represented
by
(LfMx)
2711
f(s)f (l-s)x ds.
Further, using the representation (19.4) we obtain
- XU .
(L(f*g))(x) =
e
r(i-s)rci-t) r(2-s-t)
1
du
Unir
f
*
. * ., i-s-t,
(s)g
(t)u
,.
dsdt
cr cr t s
r(i-s)rd-t) r(2-s-t)
(2ni)'
f
*
. * , , , ,,
(s)g
(t)dsdt
-xu
e
1-s-t.
u
du
cr cr
t s
r(l-s)r(l-t)f*(s)g*(t)x"2+s+tdsdt Uniy cr cr t
1 2711
s
r(s)f*(l-s)x"sds
-U 27T1
r(t)g d-t)x
dt
= (Lf)(x)(Lg)(x). This completes the proof of the Theorem 19.1.
Now
it
is
to
be
noted
that
representation
(19.4)
is
double
Mellin-Barnes type integral, its integrand involving the beta-function B(l-s,l-t). Consequently, if we replace this beta-function by ratio of two
products
of
the gamma-function,
then we
get
the
new
class
of
convolutions. For example, here we introduce the following definition.
Definition 19.1. The modified Laplace convolution of two functions f(x), g(x) e !JJ1 (L) is called the double Mellin-Barnes type integral
168
(19.17)
(rtg)(x) (f*g)(x) = - i 1 — f [r ) r^f V J J ^^T(s {2ni)2 + t) (2Hi)2
(T
Theorem 19.2. The modifedd all
f(x), g(x) € 3JT1 (L). Then
f'(s)g*(t)x—'dsdt. f*(s)g*(t)x—"dsdt.
j j r(s+t)
Laplace
convolution
x1/2(f*g)(x) € 0JT1
case
there
(19.18)
where A
for (L)
c,min(y+l/2,2^+1/2) c,min(y+1/2,2^+1/2)
c,y c,y
and in in this
(19.17) exisss
is
the following
factorization
property
(A (f»g))(x) (f*g))(x) = (A f)(x)(A g)(x), (A f)M(Ag) lx), +
is a modified
+
Laplace
+
transform
(18.1).
A
This theorem can be proved in the same way as Theorem 19.1. In the following §20 we shall prove more general result. We note that the beta-function B(s,t) = f p ^ /^g^^ ) - is is the the kernel kernel of of the the convolution c (18.1t
(19.17), but the kernel of the modified Laplace transform
is only
onb gamma-functiono
Cohsequenfiy,
here we see ohe
Ao(low1)g correspondence between (19.17) ans (19.18) Ao(low1ng
T(s+t)
<
>
(A+(f*g))
T(s) T(s)
<
>
(A f)
r(t)
<
>
(A + g ).
+
This
simple
consideration
allows
us
to
generalize
convolution
(19.17), where any gamma-function is replaced by the kernels $(x) of H-transform (see (16.8) and (16.9)). This idea initiates new trend in the convolution theory and here we give rather finished results of this class of convolutions. 169
§20.
General integral convolution: definition, existence and factorization property
The main object of the present Chapter is the general convolution which is defined as follows Definition 20.1. Let, as usually, for k=l,2,3 denote
in
ii
w
T-tr(^+b; T]rfr(i-«ik,-!k (20.1)
v
j=l
$ (T)
a.
also ,b.
the be
j=l
^
_
k,
rfr(a; +a-'x)rfr(i-P; -bfx) j=n +i
Let
;
k,
k
complex satisfy
v
numbers the
;
(k)
a.
(k)
, /3
following
J
+iK
j=m
and
the
conditions
real
of
the
positive poles
supposition for the line cr = (1/2-ico, 1/2+ioo) (see Definition
(
Re(/3U))
+
b ( k > 2 > 0,
j
j
(20.2)
k
tfe(a(k))-a(k>2
1 -
> 0,
k = 1,2,3; k = 1,2;
> 0,
j = 1,2,...,n , j = n +1,...,p ,
j
3
b(3)
Re((B\
separation 10.1)
k
> 0,
tfe(a(3)) + a ( 3 ) j
j = 1,2,....n ,
j
Re(oci3))
1 -
1,2,...,m ,
j
numbers
> 0,
*3
j = m 3 + l,,
Then the general convolution of two functions f(x), g(x) e UJT-1 (L) called the next double Mellin-Barnes type integral
(20.3)
(f*g)(x) =
1
r $ (s) $ ( t ) 1
2
$ (s+t)
(ZniV
3
cr cr
t s
170
f (s)g (t)x" s
dsdt.
is
First we establish the conditions for the existence of convolution (20.3). According to (16.11) and (16.12) denote
m (20.4)
K
~
k
n
k
u(k) +
) b
2
j=i
Pk
(
(20.5)
„
/u = Re
„
k
; a
p (k)
„
->
j=i
a
(k)
, (k)
b
q
)
j=i
k
Y
j=m +i k
p k ^
„ ^(k)
„
-
j=n +i k
J=i
qk (k)
q
k
*k
- v b
j
j=i
p - Mq
k „ , (k)
(k)
a
k
j=i J
k = 1,2,3.
Then from the asymptotic estimation for the gamma-function (4.2) we obtain
$ (s) $ (t) 1
2
exp ir k
| s+t | -K |S|-K lt|
$ (s+t) 3
(20.6)
-v
M
Is+t | 3 | s | where E = const, s,t
a
-y |t|
2
,
> oo, (s,t) e (cr x(r ) and s
(
(20.7)
P
t
q
3
~
(3)
j=l
3
„ , (3) j=l
Note here that (20.6) can be followed immediately from the asymptotic estimation (11.3).
Theorem f(x) e m'1
20.1.
The
general
(L), g(x) € m'1 1 1
convolution
(L) if 2
2
171
(20.3)
exists
for
(
2 s g n ( c +K - K ) + sgn(3r + ji -JJL ) ^ 0,
1
k
k
3
k
k
k = 1,2;
3
2 s g n ( c +c +K +/c ) + sgnt^r +^r +JLI +JI ) ^ 0,
(20.8)
s g n ( c +K - K ) + s g n ( c +/c - K ) + s g n ( c +c +K +K ] &
1
1
&
3
2
+ 2sgn(yi+r2+fii+^2--fi3)
1/2
In this case defined
as
1
2
1
2
—1
the
pair
(c ,^r ) is 3
3
follows
'
1
1
3
1
1
),
if
3
1 if
1 *1 * 3
2 ^2
I VVWV''
2
2
if
^3
C +K 1 1
C +K . 2 2
T h i s theorem can be p r o v e d by a n a l o g y w i t h Theorem 1 9 . 1 f o r
Laplace convolution. convolution
1
C +/C < C +K ; 2 2 1 1
(c +K - K , min{^ +U. -u , ? +u -u 1 1 3
Proof.
c +K < c +*c :
3
(C +K ~K , 3T +/J. -Lt ) 2 2 3 °2 * 2 ^ 3
(c , y ) 3
(20.10)
&
3
5: 0.
(f*g)(x) 6 DTI (L), where Vr3
x
( (c +/c -K. , 3r +U. -u
(20.9)
2
In f a c t ,
(f*g)(x) e x i s t s
|s+t|
3
Mt1
|s|
2
from
(20.3)
and
(20.6)
we h a v e
that
the
if
exp^/c3|s+t|-Ki|s|-K2|t|jlf*(s)g*(t)
€ L(cr xcr ) . s
t
1
S i n c e f ( x ) € DJf ( L ) , g ( x ) e JJJf1 (L), c ,7 c ,? 1 1
(20.11)
f (s) e
2
Isl
ii
i.e.
2
€ L(crs ), °g (t) e
*
Itl
ii
2
€ L(crt ),
therefore relation (20.10) holds valid if
sup
{|s+t|R3
1 "l
2 ^2
(s,t)G (
s
t -(C +K ) t 2 2 ' '
< +00.
172
'['h
exp|7T|K I S + t I -(C +K )|s|
But this is equivalent to conditions (20.8) by using Lemma 11.4.
Let x1/2(f*g)(x) 6 JJf1 (L) for some pair (c,y). Then from (20.3) we c,7
1/2
represent x
(20.13)
(f*g)(x) as follows
x"TF(x)
x1/2(f*g)(x) = -2niU to
$
dT
(T+1/2) 3
where
(20.14)
F(T) =
$ (x-t + 1/2)* (t)f (x-t + l/2)g (t)dt for T 6 (r .
2ni
1
2
T
1/2
Hence, according to Definition 15.2 x
-1
(f*g)(x) € !JJ1
(L) if and only
if (20.15)
g H/2-HT)
e7rC
'X||Tr
G
L(l/2-ioo,l/2+ico).
3
Using the equivalent symbol = of simultaneous convergence or divergence of two integrals from (20.13) and (20.14) we have
F(T) (1/2+T)
$
7TC|T| I | y ,
e
x
dx
3
$ (s)$ (t) 1
2?r
2
71c | s+t-l/2|
$ (s+t)
s+t-1/2
f*(s)g*(t)dsdt
3 t s
| S+ t |
|s|
It I
^ e x p 71 ( c + K 3 ) | s + t | - K j s | - K
( T CT
t
s
x |f (s)g ( t ) d s d t |
173
2
|t
)]
Since
(L), g(x) e Ml"1
f(x) e Ml '
(L), then
the
last
integral
converges if
(
sup
(s,t)€(CT xtr ) 1v s
s+tl
-V M 2
-M
Is
111
r r exp 71 (C+K ) |s+t I
t
(20.16)
' ( C l + I C l ) l S l " ( C 2 + l C 2 ) l t l ) ] } < + £ °Again using Lemma 11.4 we get that (20.16) holds valid if
f
2sgn(c+K - c - K ) + sgn(3r+/i -y
1
3
k
k
3
- J I )< 0, k
k = 1,2;
k
2sgn(-c i -C 2 -K i -K 2 ) + sgn(-'yi-2r2-fii-M2) * 0, (20.17) sgn(c+K -c -K ) + sgn(c+K -c -K ) +sgn(-c -c -K -K ) & 3 1 1 3 2 2 ° 1 2 1 2 + 2sgn(^+M3~^1-?'2-fi1-^2) s 0.
Now it is easy to verify that the last set of conditions (20.17) holds valid for the pair (c.^r) = (c ,v ), which is defined by (20.9). It means 1/2
that x
-1
(f*g)(x) e UJ1
(L) (here we also use (20.8) as the existence
conditions for the convolution (20.3)). Theorem 20.1 is thus completely proved.
■
In §16
we defined the H-transform of function f(x), belonging to
!JJ1 (L). Here we give definition of this transform of function f(x), such that x A f(x) € 3n_1(L)f A 6 (R.
Definition 20.2. Let A be a real constant and a function x f(x) € JH (L),i.e. it is represented in the next form
(20.18)
x A f(x)
1 2ni
f*(x+A) x"TdT,
where f (T+A) e L(cr ). Then the H-transform with the kernel $ ( T ) of the function f(x) is interpreted as follows 174
$(x+A)f*(x+A)x~Tdx = - J^ U U 2ni
(Hf)(x) = -%-A ^j 2ni
(20.19)
$(x)f*(x)x~Tdx.
Re(T)=
— +X 2 2
We n o t e t h a t i f $ ( x ) f **x) i s an a n a l y t i c f u n c t i o n i n some s t r i p
8,
invoWeino t h tthat v e r tif i c a$(x)f l l i n e(T) s fti(xa 1/2 and function J?e(x) = 1/2 + A, strip moreover We note is ann analytic in some 8, thvo vingtion vs i n t e g r lines l b l e on any n v1/2 e r t i c aand n J?e(x) l e n e o=f 1/2 th++ A, s t mmoreover i p 8 and invoWeino tht vertical fti(xa t(x)ff(xc x € 8e on |l/n(x)| thvo vingtionn ivs for integrlble any vertican t(x)ff(xc $he $he the the
ni
lene of
th+
stmip 6 and
for T € 9e |l/n(x)|
^ oo then from hauchy ipeorem ^ oo then from hauchv ipeorem f-transform (20.19) ,oincides wita ,ht H-transform defined by f-transform (20.19) ,oincides wita ,ht H-transform defined by 8) whose integral concour id the ithe
(
According According result. result.
to Definitions to Definitions
Theorem 20.2. Let kerness
2
(20.20) (20.20)
20.2 we 20.2 we
on
the
k
1
g ( x ) € JJf V
k
k
convolution
(L) if
y
following following
(H2g)(x) with UK'1
spaces
c ,y l1 i1
the
(L) and
and only
k = 1,2.
*k
( 2 0 . 3 ) exists
for
all
f(x)
€ JJJf1
(L),
if
2 f
2sgn(c +K -K ) + sgn(y + n - / )
\
sgn(c +K -K ) + sgn(c +K -K ) + sgn(c +c +K +K )+
&
[[
k
1
I
k
1
3
k
3
2
*k
2
^3
£ 0, k = 1,2; 2: 1
3
andt and,
2
1
2
+ 22sgn(r -^) + s g n ( r i +1y+r ++fi M 2 -2M 3 ) ** o. 0. 2 + (2x^i 1
(L), where In this case x1/2(f*g)(x) € JJf JJT11 (L), Vy3 (20.9),
the the
iie.
2 s &&g n ( c +/c +*c ) + s g n ( y +n ) 2: 2: 0, 0,
Then the general
obtain obtain
(H^Mx) and (H^)(x)
exitt
$ (x), $ (x) (20.1) 1
and and
H-transforms
m'1 (L),respeceively, V°2
(20.21)
20.1 20.1
moreover,
and g ( x ) e€ Jm" and JJf11 (L), V*2
c2,r2
if then
(c ,y ) is defined 3 3
x ) e JJf 3JT11 (L) for x 1 / 2 ( f «* g )) ((x) f o r aalli i f ( x ) c.r UJf1 (L) D JJi JJf1* JJf c'* V
cr
y
(L). 3
c 3 ,r 3
175
by
JJT1 (L) e€ 9H" ci>yi
Finally, H -transform 3
there
is
(H f)(x) with
the
following
the kernel
factorization
$ (T) for
3
property
the convolution
3
(20.3) by the product
°
of two H-transforms
(Hf)(x) and (Hg)(x): 1
(20.22)
of
(f*g)(xj
2
(H (f*g))(x) = (H f)(x)(H g)(x) 3
where the left
Proof.
part
of
1
2
(20.22) is understood
by Definition
20.2.
From (20.20) it is evident that 2sgn(c +c + K + K ) + sgn(y +^r +u +u ) £ 0. &
1 2
&
1 2
1
2 'l * 2
Hence according to the set of conditions (20.8) in Theorem 20.1 we see that inequalities (20.21) together with (20.20) provide the existence of 1/2
the convolution
(20.3),
and we have x
—1
(f*g)(x) G 3J1
(L) in this 3
3
case , where (c ,y ) is defined by (20.9). 1/2
We shall show that 5JT1 (L) D JJT1
(L). Indeed, we set
-7Tc Is | -y -l-e # f (s) = e |s| ,
Then according to definition of !JJ1
all e > 0.
Consequently,
(20.15) holds for the function F ( T ) ,
176
It | 2
-r -i-e Itl 2 ,
e > 0.
(L), we have that the corresponding
from
form
-TIC
0
g (t) = e
functions f(x) and g(x) belong to 3JT1 for
-1
(f*g)(x) e 5J1 (L).
Now let there be some pair (c,y) such that x
1
(L) and HIT2
(20.13)-(20.15)
2 (L)
respectively
it follows
that
which has the following definite
1
F ( T ) = ^ij exp|-7r|c |x-t + l/2|+c |t| 11 |x-t + l/2| (20.23) x
dt.
t
In this case using the replacement x-t+1/2 = s and the equivalent symbol = of convergence or divergence of two integrals from (20.23) we obtain
F(x) $
^irclTL.y
dx
(1/2+T) 3
|s+t|
|s|
|t|
exp|7i|(c+K )|s+t|
[■('
(T (T t s -1 -£ "(C+K )ISI~(C +K ) t l l 1 1 2 2 '
IstI
Idsdtl
Now using conditions (20.20) and applying Lemma 15.3 we get that the last integral converges if and only if inequality (20.16) holds valid, i.e. we again obtain conditions (20.17). Here conditions (20.20) allow us to abbreviate (20.17) to the following conditions
2 s g n ( c - ( c +K - K )) + s g n ( ^ - ( ^ + [i -[i )} (20.24)
s g n ( c - ( c +K - K ] &
0, k = 1,2;
+ s g n ( c - ( c +K - K )) - s g n ( c +c +/c +/c )
1 1 3
2
2
3
1
2
1
2
+ 2sgn(2r-(*1+*2+fi1+^2-M3)) < 0.
Now for any pair (c,^), satisfying (20.24) we must show that JTJ
(L) D
c , 'y -1 (L), where c > „ c3 ,2"3
JT1
(c ,y ) is defined by 3
(20.9).
3
first inequalities of (20.24) it follows that 177
Indeed, from
the two
c ^ min{
But
from
(20.9) we have K3>c2+*2-K3}} 'VV+VS*}
min{c + Kmin{c + Km X1 (L) z> m'11 c , y (L) z> m'c 5JT c y
Now Now
C + K - K , 1 1 3
inequalities
3
i i
= min{c + K -K , c + K -K }} .} Hence 1 1 3
follows follows
2
from from
2 K
ii c ^
3
properties properties
((15.12 ((15.12
that that
two two
ff ii rr ss tt
(L) for all y, y e R. (L) for all y, y3 e R.
y
c3>9r3
ll ee tt
c
C + K - K } . 2 2 3
cc = = cc
3
= +/c -K - K }>.. = min{c min{c +K +K -K - K ,, cc +/c
3 3
11 11 33
22
22
Then Then
3 3
m'1 ( 2 0 . 2 4 ) we have IJJf 3JI *1 (L) D :> JJf1
from from
_
* ((L), * L ) , k = 1,2. 1,2.
+ K K. , y +LlLl LlLI cc k +Kk K3 , y k + * k 3
cc ,,yy
1 1 Consequently, according to (20.9) we must prove t h a t m lUT (L) z> UTf1
c ,y 3 3
c,r c,y
in the case c +/c = c + K , i.e. , when in the case c +/c = c + K , i.e. , when 1 1
1 1
2 2
2 2
(L)
c = c = c + K -K = c + K -K . In c = c = c + K -K = c + K -K . In 3 3
1 1
this this case case from from the the last last inequality inequality of of (20.24) (20.24) we we get get
1 1
3 3
2 2
2 2
3 3
2 s &g n ( y - ( y +y +fi +JLI - / ) ) < tos g n ( c +c +K +K ) < 1, &
11
to
22 **1 1 **2 2 *^3 3
1 1 2 2 11
y y +y +LI +M +LI +jn -LI\ -jn*. Hence H e n c e Mf UJT11 (L) (L) 3 3 3 IJJf1 y ± ^ y 11 *
i . ee..
C
'*
°3
2
(L) for f o r any a n y pair p a i r (c,y), (c,y), (L) 3
satisfying (20.24). Further, as it is shown above, we have x
(f*g)(x) <= JT Jl c
Using
representations
(20.13)-(20.14) (20.13)-(20.14J
for
the convolution
(L). 3>3r3
(f*g3(x),
Usingding ro Definition (202 we obtain (H3(f*g))Hx) =
tio/2
r
^nT
U ( x + 1 / 2 ) $ (TIi/2)x"Tdx
J
cr
-
T
= ^ _JL _ _2 f\ $
(2*i) J '' (2ni)2JJcr J cr
T T1/2 -t+1/2)* (t)f*(x-t+l/2)g*(t)x__1/2 dtdT (T-t+1/2)* (t)f*(x-t+l/2)g*(t)xdtdT
cr cr Tx tt
=
^ _ [ $ (s)f'(s)$ (t)g*(t)x-s-tdsdt (27ii) 2 J J 1 2 cr cr s s t t
= (H f)(x)(H g)(x). 1 1
2 2
178
Here the existence of the H-transforms (Hf)(x) and (Hg)(x) is provided by inequalities (20.20). Theorem 20.2 is completely proved.
We
defined
the
general
convolution
as
double
■
contour
integral,
however, under some additional conditions it can be represented by a real form. The following theorem is true.
Theorem 20.3. Let
the
parameters
, (3
a j
conditions
(20.2). Let
also
the
following
, a
j
satisfy
, b j
the
j
inequalities
hold
2sgn(fc - K ) + sgn(/i -JLI - 1 ) > 0, k = 1,2; k
3
*k
*3
2sgn(K + K ) + sgn(u +u -1) > 0, 1 2
(20.25)
1 2
2min-jsgn(K - K ) + sgn(/c -/c ) , sgn(/c -/c ) + sgn(/c +K ) , 3
2 3
sgn(K -K ) + sgn(
Then
the
general
convolution
1
1 3 2
/
can
(20.3)
1 2
+ sgn(jj. +M ~M ~2) > 0.
be
represented
2
integral
on R
as
follows
H[£ .5]f( U )g(v: dudv uv
(f*g)(x) =
(20.26)
o o where
H [x,y]
H [x,y]
is
H-function
of
two variables
$ (s) $
1
1
$
(2711 ]
2
(t)
-s -t ,
x
(s+t)
of
y
type
(10.1!
,.
dsdt
3 cr (T t s
p -n (20.27)
*3
, q -m ; m , n ; m , n 3
q ^3
3
»
3
p *3
1
1
2
; p >q ; p >q M
^1
^ 2
179
n
(/3(3),b(3))
2 x,y
2
,
(a
m +1 , q 3 3
(3),
(3)
,a
n +1, p 3 3
by
double
ro(3)
(/3 r
(3)
(a
,
, (3K
,b
(3)
,a
)
1 ,m 3
^
(1)
: (a
rod)
)
(/3
(1).
,a
1 , p 1
In fact, from
(2).
; (a
;fo(2)(/3
,(1K ,b )
1 , n : 3
Proof.
, (2)
)
,a
, (2K
,b
11.1
it
]
1 >P 2
)
1 ,q 1
Theorem
x
)
1 ,q 2
follows
that
the
double
integral in (20.27) converges if and only if conditions (20.25) hold. In this case it is evident, that
$ (s) $ (t) SUp (s,t)€((T X(T ) s t
1
2 (s + t )
$ 3
Hence convolution (20.3) exists for all f(x),g(x) e 9JI (L).
Further, conditions H-function
(20.27).
convolution
can be
(20.2) are required for the existence of the
Finally
representation
transformed
from
(20.26)
(20.3) by
of
the
the using
general
the double
Mellin transform of H-function (20.27)
(20.28)
m 1 H [x,y];s,t}- = 1
'
$ (s) $ (t) * ?— , (s,t) € (
(S+t)
s
t
(see also §7, Theorem 7.3). Theorem 20.3 is thus proved.
We conclude by remarking that if $ (s+t) = 1, then the convolution (20.3) is the product of H-transforms (Hf)(x) and (Hg)(x), i.e. in the present case we have
(f*g)(x) = (H f M x M H g)(x),
180
if
* (s+t) = 1.
§21.
According
Typical examples of the general convolutions
to Definition
evident that convolution
20.1
of
the general
convolution,
it
is
(20.3), generally speaking, does not possess
the commutation property, i.e. (f*g)(x) is not always equal to (g*f)(x). It is not difficult to note that (f*g)(x) = (g*f)(x) if $ (T) s $ ( T ).
In this section we consider only symmetrical cases of the general convolution, when $ (x) = $ (x) and both functions f(x) and g(x) belong to JJ]
(L). Some typical examples are also obtained. For convenience we
c, y denote $ (x) = $ (x) = <Mx) and 1 2
Theorem 21.1. The
(21.1)
(K , u ) = (K ,LI ) = 1 1 2 2
(K,U).
convolution
$(s)$(t) *
(f*g)(x)
$
2
(27ri)
(s+ t
)f
. * ., -s-t, ,.
(s)g (t)x
dsdt
3
exists
for
all
fix),
(L) if
g(x) e 3T1
and only
if
c,7
2sgn(c+K-K ) + sgn(^r+fi-|Li ) ^ 0, 2sgn(c+K) + sgn(3r+/ii) ^ 0,
(21.2)
2sgn(c+K-K ) + sgn(2y+2^-/ui ) ^ 0.
In
this
case
(f*g)(x) = (g*f)(x)
and x 1 / 2 ( f * g ) ( x )
e JJl * ( L ) , c , If 3
= C+K-K , K = mint^+^i-ji ,2y+2n-iA For convolution
(21.3)
Moreover,
(21.1) the
); K , [i factorization
are
defined property
(H3(f*g))(x) = (Hf)(x)(Hg)(x)
if 181
where
c 3
3
by (20.4),(20.7). holds
2sgn(K-K ) + sgn(/n-/n -1) > 0, 2sgn(K) + sgn(2fi-l) > 0,
(21.4)
2sgn(/c-/c ) + sgn(2/i-jLi -2) > 0,
then
the convolution
(21.1) exists
can be represented
as
(21.5)
(f*g)(x) =
for
all
f(x),
(L) and
g(x) € JH
it
follows
dudv H [[u- ,-lf(u)g(v: & vj uv
where
H [x,y]
<S>(s)$(t) -s -t , .. —=-—-, r-T- x y dsdt y $ (s+t)
1 {2niV
cr cr t s
(/3(3),b(3))
m +1 ,Hq 3 3
p -n ,q -m ; m, n; m, n 3 3 ^3 3 x,y q 3 , P 3 ; p>q;p,q
(21.6)
f
(3) ( 3 ) .
(a
,a
)n +1, p 3
(/3(3),b(3))
:(cc,a)
;(a,a)
1,m '3
(a(3),a(3))
Proof. ig taking
1,p
(|3,b)
1, n : '3
;(/3,b)
1,q
3
1 1 ,p
1,q
y
According to Theorem 20.1 of the convolution existence,
( y )) = ( y )) = (c,y) (cc ,,y (cc ,,y (c,^)
and and
(K ,JH ) = (K ,/I ) = (K,JLI)
(20.8), we get the following conditions
2sgn(c+jc-K (21.7)
) + sgn(?+/a-/j. ) ^ 0,
2sgn(c+K) + sgn(2r+/i) > 0, 2sgn(c+K-K ) + sgn(c+/c) + 2sgn(23r+2fi-fx ) > 0 .
182
from
From the two first inequalities of (21.7) it follows that C+K-K
^ 0,
C+K ^ 0. Hence in this case it is evident that the last inequality out of (21.7) is equivalent to the last one out of (21.2).
Consequently,
(21.7) can be written in simpler form than (21.2).
Now according to (20.20), the second
inequality out of
(20.21), and applying Theorem 20.2 from (21.2) it follows that the convolution
(21.1) exists for all f(x), g(x) e 3JT1 (L) if and only if the set of conditions (21.4) holds valid. Further, in accordance with (20.25) from Theorem 20.3, the double integral in (21.6) converges if and only if
2sgn(K-K ) + sgn(n-^. (21.8)
-1) > 0,
2sgn(/c) + sgn(2/n-l) > 0, 2min-{2sgn(K-K ), sgn(K-*c ) + sgn(
"}
+ sgn(2M~fi -2) > 0.
Here two first inequalities of (21.8) show, that K ^ 0, K-K £ 0. Then
*
the last inequality from (21.8) means that the condition 2fi-/Lt -2 > 0 is required if and only if K-K
Hence it is not difficult to obtain
that (21.8) is equivalent to (21.4). Thus Theorem 21.1 is proved.
Now we consider some original examples of convolution (21.1).
Example 21.1. Let <Mx) = T(x) and $ (x) = T(x)/k (x), where k (x) is Mellin transform (15.3) of some function k(x). Then the corresponding convolution can be written in the form
(21.9)
r(s)ru) * _,_.,*, x *,., -s-t, ,.
(f*g) (x) ( 27T 2* ]
r(
+t)
k (s+t)f (s)g (t)x
dsdt.
At first we propose that the function k(x) is Fox's H-function (1.5) and * k (x) is defined by formula (16.9) 183
m n "FT T O +b x) F T r(l-a -a x) j
j
p FT
j
i-l_
q r(a +a x) F T r(l-j3 -b T ) J
j
j=n+i
j
j
j=m+i
According to formulas
(20.4),
ji = 0 and
21.1
by Theorem
j
J=1
k*(T) = -^
existence of the convolution
(20.7) in our case K = 1/2,
(20.5) and
it
is not
difficult
to obtain
that
(21.9) for all f(x), g(x) e 3JT
the
(L) is
* * provided by the next conditions (here K ,a for $ (x)=r(x)/k (x)) 3
3
3
* 2sgn(c+l/2-K ) + sgn(y-ju ) £ 0,
(2i.io)
3
3
;
^
2sgn(c+l/2-K ) + sgn(2?-^F) * 0.
Further the factorization property (21.3) in this case can be obtained by the aid of composition structure of the corresponding H-transforms. According
to
formula
(18.1)
(Hf)(x) = (Af)(x)
and
since
$ (x) = 1
T(x)/k*(x) then by the Theorems 16.4 and 16.5 (H f)(x) = (A oK"" ) (f (x)) , where the H-transform (K f)(x) is the inverse transform for (Kf)(x) * with kernel k (x). Hence the corresponding formula (21.3) takes the next form
(H (f*g) )(x) = ((A o K 1 )(f*g) )(x) = (A f)(x)(A g)(x).
(21.11)
3
From conditions existence
of
real
+
+
+
+
(21.4) it is not difficult representation
(21.5)
it
is
+
to see that for sufficient
that
the the
following inequality be valid
(21.12)
2sgn(l/2-K ) - sgn(fx +2) > 0.
However, in this case the H-function of two variables
(21.6) can be
evaluated by the reduction property (12.3), formula (20.28) and the next identity (see
Nguyen Thanh Hai and S.B.Yakubovich (1990)) 184
r(s)nt) * r(s+t) l
(21.13)
, , J
^ s-l
k(x+y)x
t-1 , .
y
dxdy.
o o
«N -
Hence H|^,^| in (21.5) is equal to k
and we obtain the so-
called sum-convolution
(21.14)
The
k X
[ ^]f(u)g(V
(f*g) (x) =
identity
(21.13)
is
true
dudv uv
if k(x) € L(0,oo),
therefore, we
can
consider the convolution (21.9) in form (21.5) for arbitrary kernel k(x) € L(0,oo). For this convolution it not difficult -1/2 to get the norms estimations in L-spaces. Thus, if f(x), g(x) € L(x
;0,oo) from the following chain
of inequalities 00
03 00
dx
*H?] f(u:
dudv g(v)
uv ,_ . , .,dudv — — f ( u t) g ( v )
|k(x)|dx
i4-\/
I
°
rig(v)
rlf(u) |k(x)|dx
I in;
dv,
-du Vu
Vv
we have
(21.15)
|(f*g)J
^ M|fI L(0,oo)
where M
_x L(x
;0, oo)
m L(x
;0, oo) '
|k(x)Idx.
Many examples of the sum-convolutions can be obtained by the Table 13.1. Let k(x) = 2K (ZVx~) o
is Macdonald function 185
(see line 47 in the
Table 13.1). Then the corresponding sum-convolution (21.14) and factori zation property (21.11) take a form
(f*g)+(x) = 2 (21.16)
K
of*■ 2 /^vl' f ( u ) g ( v )uv ^X •
(A^tfsg) )(x) = (A f)(x)(A g)(x).
1 Example 21.2. If $(x) = 1/T, $ (x) = — , then similarly from 3 xk (x) the representation (21.9) we obtain the so-called max-convolution
(21.17)
(f*g)
k x max{u" ,v~ > f(u)g(v vdudv
(x)
It is not difficult to see the corresponding conditions of existence from (21.2), (21.4) by K = 0, JI = 1. The H-function (21.6) in this case can be evaluated by the maximum reduction property (12.4) and it is equal to k x max{u ,v > . Further, the factorization property (21.3) takes the next form (H (f*g)
)(x)
3
(21.18)
= ((I x_1)o K _1 )(f*g)
)(x) = I1(x"1f(x))I1(x"1g(x)),
max
-
-
where the simplest operator I (x"f(x)) is defined by Table 17.1 (line 7).
As — 1/2
L(x
in case of sum-convolution, if k(x) € L(0,oo), f(x), g(x) € ;0,oo) then for max-convolution (21.17) the norms estimation is
also true
(21.19)
||(f*g)oaJ
*2M|f| L(0,oo)
L(x
where constant M is defined by (21.15). 186
|g| ;0,oo)
L(x
;0,oo)'
Actually, 00 00
CO CO CO CO
kfx max{u" 1 ,v" 1 }]f(u)g(v)^X dx II JJ uv 00 00 0
oo oo
**
oo oo oo oo
|k(x)|dx
min{u,v>|f(u)g(v)|^X min{u,v>|f(u)g(v)|^^
00
00 0
oo oo
oo oo oo oo
==
|k(x)|dx |k(x)|dx ' '
1
o
<<
rif JJ
oo o
|k(x)|dx |k(x)|dx
00
— — ^^ -|f(u)g(v)|^X && max{u,v>' '' uv uv max{u,v>'
nrif(u)i f(u)i
rig(v)i 7ig(v)i
du du
dv. dv.
J J vv ^^ JJ v'v" v'v" 00
0 0
According to Table 13.1 (line 48) the corresponding max-convolution (21.17) with Macdonald function K (2Vx~) (2Vx"~) is the following o oo 00
(21.20)
(x) (( ff *«gg))r M o aX x(x>
= 4z\ =
oo 00
1
|[ K K Oo(( 22 // xX mma ax x< {u u- - ,1v, -v1- > 1 )>f)(fu()ug)(gv()v^)X^ ..
00 0
The basic factorization property (21.18) for this convolution takes the the form
(21.21)
(2Vx)\ 1o[(f*g) °|"(f*g) (u)l J2x1/2K (2Vx)\" (u)l I III max JJ 1 I max
=
( x - 1 f(x)) ( x ) ) II1^ (x^gtx)), (x-1g(x)) II1 ^(x'V -
1/2 where operator of the inverse Meijer transform JJ2x KK i(2i/x)l 2 x1/2 ( 2 v ^ ) r 1 o°|f(u)| rf(u)l
is defined by Table 17.1 (line 25). 187
Now we give the table of the kernels for sum- and max-convolutions (21.14), (21.17) and the corresponding H-transforms in the left parts of the factorization equalities (21.11), (21.18). This table is based on the Table 13.1.
TABLE 21.1
X+y kf(max(x,y)J ]
1
2
(H3f)(x)
e-x-y
E (x A ' V 1 )
exp - m a x ( x , y )
+
3
r(a)(l+x+y)"a
4
T(a)|l+max(x,y
5
nsy
(1 x
- -y } +
6
f
7
Eij-x-yj
8
9
^y(l-max(x,y
(
(x
>r
|r(l+a)x(l+x)"1"a|
l
(x A x
)
+ , a i - a -lx
(x TI_
•):'
x
)
[■XT' [IV^IA;1)
Ei - m a x ( x , y )
Vn
- a A - i a. A x )
l1/2x-l/z)
(x
erfc{Vx+y)
t 1/2.-1 -1/2N
10
Vn e r f c ( ^ m a x ( xTyT)
(x
11
T(v,x+y)
(x I
12
f[v,max(x,y)]
(x A x
A x
X
+
188
)
+
) )
13
f-'-^r2]/,
J42V^)
x (l+v/2;l+v;~)|
14
( ,
^
{x—r[--^-] iF2 ( 1+ , /2;
Jv^max(xfy)J
2+v/2,l+v;~)|
|x- 1/2 e- x/2 W
15
16
^
i
■■[:]/,[: H
18
■"['] ,Ft[i I""""* "]
20
r[V] /,[•;'
J
(iV 1 )* J2K^(2^)|
2 K^ 2^max(x,y)
17
19
(x)V 1 1/2,IV2
(x
1—a T a-b b - 1 x I X ) +
H^V 1 ^ 1 * 13 ^} 1 f e -l/(2.) (l)!"1 | 1-c,(a-b)/2 X J
-x-y
|xd + x) c - a - b - 1 r[ 1 ^; 1 + b ]
r[V]/JV | -max(x »]
x F (c-a,c-b;l+c;-x)\ 2 1 J -l
21
r(a)T(a-b+l)
22
r(a)r(a-b+l)
1
tf|a,b;x+yl
b
b
a
1
|r(l+a-b)x " (l+x) ~ " j
tf|a,b;max(x,y)l
189
|xr(l+a)r(a-b+l)*(l+a,l+b;x)i
§22. Case of the same kernels: the general Laplace convolution
In this section putting $ (x) = $ (T) = $ (x) = <Mx) in Definition 20.1
we obtain
corresponding convolution
the so-called
algebraic
classical
properties.
is the modified
case
of convolution
The simplest
Laplace
convolution
example
(19.17).
difficult to show from the Theorem 20.3 that the integral convolution
(19.17) diverges. But we can bring
with
of such It is
not
(20.27) for
the modified
Laplace
convolution to the similar form (19.1). Our new approach to convolution constructions naturally allows to generalize convolution
and obtain
I.H.Dimovski
the known examples
and V.S.Kiryakova
the classical notion of (see I.H.Dimovski
(1984,1985),
V.S.Kiryakova
(1974),
(1989)) as
well as the new ones with the corresponding factorization properties. Earlier these constructions were used in operational calculus. The known ways of constructing
the operational
theory are also applied
to
the
objects considered. Now we establish the main theorem for convolution (20.3) in the case of
the same
simplified.
We
kernels.
The corresponding
denote
(K ,/LX ) = (K^,/^J U 2'fl2)
conditions
accordance with (20.4), (20.5) m [j=i
n
p
q
j=i
j=n+i
j=m+i
P
q
j=i
j=i
p
Theorem 22.1. The
(22.1)
(f*g)(x) =
J
q
p - q
Ea - Eb
fi = Re
convolution
$(s)$(t) »(s+t)
1
(2rrir (T (T t s
where
190
* f
are greatly
= (K^,/I„) = (K,/I), = U 3'^3) = ( K ^ ] '
. *
, - s - t , ,. d s d t '
( s ) g U ) X
where
in
f r r ( V V ) rTr(i-aj-ajT) (22.2)
<Mx) = 7 7 rfa.+a.x] 7 7 rfl-p.-b.r)
and the parameters
satisfying
,
the following
Re{(B ) + b / 2 > 0, J J
j = 1 , 2 , . . . ,m
l-/te(a
j = 1,2,. . . , n
) - a J
(22.3) j
j
■ l-/?e(3 ) - b
for
all
fix),
g ( x ) e 5TI
> 0, j
Ke(a ) + a
exists
conditions
> 0,
j = n + 1 , .. . , p
> 0,
(L) if
j = m + 1 , .• -q
and only
[ 2sgn(c) + sgn(y-
P q Ea - £ b
-
j=i
P
q
J=l
J=l
E a - Ib
2 s g n ( c ) + sgn(2r+M~2
1/2
x
(22.5)
For convolution (22.6)
o,
j=i
2sgn(c+K) + s g n ( r + / - t ) - 0 ,
(22.4)
i n t h i s case
if
-1
*
( f * g ) ( x ) € !JJ1
".
) * 0.
, (D> where y
=
2
= min(2r-jm#,2y+jLi-jLi#) and
p q [ a - [ b
(22.1) the factorization
property
(H(f*g))(x) = (Hf)(x)(Hg)(x).
Moreover, if 191
holds
2sgn(K) + sgn(2fi-l) > 0, (22.7) fi# + 1 < min(0,Li-l),
then
the
convolution
can be represented
(22.1) exists as
all
(L) and
f(x), g(x) e 5J1
H[n >^u)g(v
(f*g)(x) =
(22.8)
for
it
follows
dudv uv
where
*(s)*(t) -s -t , ,, —^ ^j— x y dsdt y $(s+t)
1
H[x,y] =
Uni)'
cr cr t s
(22.9)
O.b) p-n,q-m;m,n;m,n
=H
q ,P
,(/3,b) m+l,q
: (a,a)
1,m
,(a,a)
n+l , p
: O.b)
;0,b)
1, n
1 ,q
1
Proof. According to * (K , a ) = ( K , U ) , LI = Lt+Lt , 3
1, P
x,y
;p,q;p,q
(a,a)
3 3
)
;(a,a)
1,p
Theorem where ix
1
1, q
2
2
21.1 taking (K ,U ) = (K ,U ) = i s d e f i n e d by ( 2 2 . 5 ) i t i s n o t
*
*
difficult to obtain from conditions (21.2) the equivalent inequalities (22.4). Further, since K in (21.4) hold
= K, then the first and the last inequalities
if fi#+l < 0 and LI-LI#-2 > 0 and we get
(22.7). Thus Theorem 22.1 is established.
Remark
22.1. For
modified
Laplace
LI = 0, LI = -1/2 and we obtain only
the conditions
■
convolution
the conditions
(19.17) K = 1/2, (22.4) for
this
convolution (the conditions (22.7) do not hold and the integral (22.9) diverges). Now we consider for convolution (22.1) some approach of constructing the integral representation as a generalization of the classical Laplace integral
(19.1) which
is based
on
192
the composition
structure
of
the
H-transform r
,
. s
(see §16). As we noted above, if we set <Ms) = r(s), then
= B(s,t) is the classical beta-function and the corresponding
convolution (22.1) is the modified Laplace convolution.
s Definition 22.1. We shall call the kernel *[$(s+t) f * * f ^ in (22.1) as the
general beta-function and denote it by B(s,t). The convolution (22.1) is naturally called the general Laplace convolution.
Next, we use the value of the beta-integral
(22.10)
B(s,t) =
u
(1-u) du
and we represent the general beta-function B(s,t) as follows
\~\
r|2|5.+b.(s+t)|
7^"r|2-2a.-a.(s+t)] 1 *(s+t)
8(s,t)= "TTrfa +a s l r f a +a t
T T r [ l - | 3 -b s | r f l - 0 -b t |
x F T BI/3 +b s,/3 +b t) "TTBIl-a -a s , l - a -a t (22.11) "PJ r [ 2 j 3 . + b . ( s + t ) |
"f7r|2-2a.-a.(s+t)j 1 <Ms+t)
T T r f a +a s l r f a +a t ]
T~T r f i - 3 -b s ] r f l - / 3 -b 1
I
m
FT
f £.+b .s-i
I
0.+b . t-i (1-u ) j
n
^. FT
-a -a s j
j
(1-u )
j
j
du
j=l
We note that all beta-integrals in (22.11) converge by the conditions (22.3). The following theorem is true. 193
Theorem
(22.3)-(22.4) hold
(22.12)
Let
22.2.
fix),
valid.
and
g(x) G ffl (L)
Moreover, p-n
q-m
j=l
J=l
if
the
next
the
inequality
conditions takes
place
)+sgn(y-ju) £ 0,
2sgn(c-
where q-m
p-n
q-m
H = Re
p-n
j=i
t h e n the is
next
q-m-p+n 2
j;b-[a
representation
of
j=i
the general
Laplace
convolution
(22.1)
true
,
(f*g)(x)
=
f(/3,b) ,(/3,b) m+1 q HP"n'q"m ' q P ^ ' I (a,a) ,(a,a) v
-TT ° ^ °
n+1 , p
u^Of 1 A" V l (2,3 , b ) ° ^ °
m ,.(m+n)..
J
(1-u )
J
1,n
u0 H
, l ( |(2a - l , a j 1,0X
(3 - 1
T"T Tu J!_= l
)
J
J
j
n
-l n -a -a F T v J(l-v ) j j j! = i
(22.13) x H
x H
where
sign
]
0,0 p-n,q-m
0,0 p-n,q-m
((a,a) O.b
n+l , p m+l , q
f ( a , aa )-*
|"o denotes
• |f(y
I
n+l,p
(0 b) m+l
the
, qy
r
g(y) L
composition
J= i
dv =
T T u i J!
TvjaJ
j=i
m n xj~7(l-u.)"bjf7(l-v.)aj j=i
J
j=i
of m H-transforms,
dudv
J
du = ]
[du.
j= i
||dv.. J=i
x
j=i
y
J
194
Proof. Since the conditions (22.3)-(22.5) hold then the convolution (22.1) exists
and
according
to
property
(23.14) and
representation
(20.13) we can obtain that our convolution has the next form
f(/3,b) (22.14)
p-n,q-m q»p
(f*g)(x)
,(/3,b) m + l,q
(a,a)
Th(u)l,
1 ,m
, (a,a)
n+l , p
1 ,n
where H-transform in (22.14) is defined by (16.18) and h(u) is equal to
(22.15)
h(u) =
^
F(T)U
dx ,
and
(22.16)
F(T) =
*(x-t + l/2)*(t)f (x-t + l/2)g (t)dt
2ni
for T 6 (T
Further we use the composition properties of H-transform
and by the
Theorem 16.7 and property (23.14) it can be established that h(u) is the composition chain of m+n H-transforms
in forms
(16.37),
(16.38) of a
function h (u) which is equal to
T T B||3 +b s f £ +b t ] " T T B l l - a - a s , l - a - a
h (u) = 1
(27ri) 2
\=[
{ J
J
J J \= [ I
J
J J
t|
J J J
t s (22.17) f*(s)g*(t)u
s
t
dsdt
fTrfa + as) 77rfi-^ -b s) r r r k + a t ] TT rfi-P.-bt) j=n+i^
;
j=m+i ^
J
J
Hence in accordance with condition
'j=n+i ^
J
J ;
J
j = m+i ^
J
J
(22.12) and Theorem 16.3 the next
H-transform of functions f (x) and g(x) exists in 3J1
(L)
c,7
0,0 H *p-n,q-m
n+i, p
f (s)
■[f(y)](x) = ^ j
x
ds
O.b)m+l , qs
195
j=n+iv
J
j=m+i
v
y
Finally, since each beta-function in (22.17) is bounded if the condi tions (22.3) take place then double integral
(22.17) converges. Using
the beta-integral
it is not difficult
(22.10) and Fubini
obtain now the representation 22.2 is proved.
theorem
(22.13) of convolution
(22.1).
to
Theorem
■
Remark 22.2. The equality (22.13) can be written in terms of the other H- or G-transforms in depending on the position of corresponding gamma-functions (see composition Theorem 16.7). Now we consider some examples of general Laplace convolution in case of G-transforms and their representations of type (22.13).
Example a
22.1. Let
in
(22.2) p = n, q = m, b. = 1,
j = l,...,m,
= 1 , j = l , . . . , n . Then we have the convolution (22.1) for G-transform
(17.1) with the kernel
*(T, = rTr(p.+x) rrr(i-«-,
(22.18)
v
j=i
y
j=i
v
The factorization equality (22.6) by the corresponding conditions in the Theorem 22.1 takes the form
„m,n n,m
(
f (a)
(a)
'■■]•[ (f*g)(u)
(0) l,m
J
L
(x)
^m,n = G
n,m
(0)
::: K f,u, i
(X)
(22.19) x Gm,n n,m
{ ^)
)
>[g(u)l(x).
1 ,n '
(0)1,m We represent the general beta-function (22.11) as follows
-A- T(20s +s+t) -A- T(2-2a -s-t)
B(s.t) = > I J 1
(22.20)
II ^ r(0.+s+t) J 1 rd-a.-s-t)
x F T B 0 +b s,0 +b t] "fTB|l-a -a s,l-a -a t|
196
Hence the analog of representation (22.13) can be constructed with the composition of m+n integro-differential operators which are defined by the Mellin correspondences 8,9 from Table 18.2 T(a+s) < TO+s)
a /3-a -(3 > x TI x
T(a-s) <
>x
i-6TS-a a-i
I
x
ro-s) Theorem 22.3. Let f(x), g(x) e !JJf (22.4) hold
valid.
convolution
(22.1) with
Then the next the kernel
m in
, 2<sp / 3 . --pp \ /
(f*g)(x) = "["Jo
X
j
j
I
(L) and the conditions
representation (22.20) is \
-p j
X
j=l V
J
j=1
i-^u. a -l i-2a j
• U. X
Laplace
true
, ja
II n
of"[o
of the general
(22.3)-
CC -1
I J
X
I
1
f m /Sj -l _
0
^ -l n
-a
-a
(l-u ) j F T v j(i-v ) j
.. FTu
(22.21)
J
J
_
J
J
0 m+n
X f
(
m 111
n 11
\
ill
/
II
) dudv
xTTCTTv. g xrid-u j^fTd-v : j=l
j=l
J
J
V j
= 1
j=i
Remark 22.3. Using the results of this section we can get many examples of convolution (22.1) and their composition representations. For classical G-transforms these examples will be considered in §24. Finally we note the associative property for convolution (22.1). For this we must define the repeated convolution (f*(g*h))(x).
Definition 22.2. Let f(x), g(x), h(x) e Til
(L). Then we shall
understand the relation (f*(g*h))(x) as the following equality (22.22)
x 1/2 (f*(g*h))(x) = (f*(x1/2(g*h)(x)))(x).
197
Supposing
the
validity
of
the
Theorem
22.1
from
factorization
property (22.6) and Definition 20.2 it follows that
(H(f*(g*h)))(x) = x" 1/2 (x 1/2 (Hf)(x)(H(g*h))(x)) = x" 1/2 (x 1/2 (Hf)(x)x" 1/2 (x 1/2 (Hg)(x)(Hh)(x))) = (Hf)(x)(Hg)(x)(Hh)(x) = x" 1/2 (x 1/2 (Hh)(x)x" 1/2 (x 1/2 (Hf)(x)
(Hg)(x)))
= (H((f*g)*h))(x).
Hence using the unity of H-transform
in space Ufi
(L) we obtain the
associative property of convolution (22.1)
(f*(g*h))(x) = ((f*g)*h)(x).
§23. G-convolution and its typical examples
The general convolution (20.3) can be simplified in the same way as in §13, when it was introduced the G-function of two variables as a (k )
special case of the H-function (10.1) with all coefficients a
(k)
,b j
j
(k = 1,2,3) equal to 1. In this section we will
consider
this simplification
(so-called
G-convolution) according to §20. The theorems of existence, representa tions and factorization property for the corresponding G-transforms will be established. examples
of
Moreover,
the general
it
is not difficult
convolution
to see
(20.3) contain
the
that many of elements
of
G-function of two variables (13.1). Therefore the notion of G- convolu tion requires independent consideration.
198
Definition 23.1. As in (13.2), for k = 1,2,3 we denote m
n k
k
(k)
]~7 ro (23.1)
* (x)
+x) J~T r(i-cx(k)-T)
j=l P
k
JfJ; °I
r V k ) + r ) f j r(i-/3
77 j=n +i k
Let also
j = m +i k
, (3
the complex numbers a j
satisfy
the following
j
!
of the poles separation supposition for the line (r = ( —-ioo,
Reipik))
+— j
— - K e ( a ( k ) ) > 0,
j = 1,2,..
j
ReUi3)
(23.2)
3
{3)
Re{a )
j = n +1, .
+ 1 > 0,
j
3
{3)
Re((B )
j = m +1, .
< 0,
j
3
type
Vs) V
1
(f»g)(x) =
the
P
3
••v
integral
\
(27ii)
Since
■ • '
o f t w o f u n c t i o n s f ( x ) , g ( x ) e !JJ1 (L) is c a l l e d t h e
next double Mellin-Barnes
(23.3)
k = 1,2;
k
j = 1 , 2 , . . . ,n ;
) < 0,
j
Then the G-convolution
—+ioo)
j = 1 , 2 , . . . , mk , k = 1 , 2 , 3
> 0,
2
2
conditions
!
functions
(s+
U f
t)
*( s ) g *
-s-t
Ct)xBtdsdt.
3
^ (x),
k = 1,2,3
are
the
kernels
of
k
corresponding G-transforms (17.1) then from Definition 17.2 we denote
*
K
k
M
k
(23.4)
c
(23.5)
•:■* Z-!"-Z»r
k
= m
k
+ n
k
j =l
-— » — 2
j =i
199
k = 1,2,3.
the
From
(20.4),
(20.5),
(20.7)
we have K = c , f i = y , k k
k
k
= 1,2,3,
and
k
P 3- C I 3 *
*
"3 = V ^
"V
Hence,in accordance with Theorem 20.1-20.3 we get the following results.
Theorem 23.1. The G-convolution g(x) € 3JT1
(23.3) exists
for
f(x) e !Jfl *
(L),
(L) if
2
2
f 1
2sgn(c +c -c ) + sgn(y +y -y ) ^ 0, & e k
k
3
k
k
k = 1,2;
3
* * * * 2sgn(c +c +c +c ) + sgn(y +y +y +^ ) > 0, * * * * * * sgn(c +c -c ) + sgn(c +c -c ) + sgn(c +c +c +c )
(23.6)
6
1
1
&
3
2
2
e
3
1
2
1
2
* * + 2sgn(y i +y 2 +* i +r 2 -y 3 ) * 0. 1/2
In this case
x
—1
(f*g)(x) <= 3J1c
defined
as
(L), where
,y 3
the
pair
(c ,y ) is
3
follows
[ (c i+ c*-c*,
VV*3}'
if
c +c < c +c ; 1 1
(c +c - c , y +y - y (23.7)
2
(c ,y ) = i 3
3
!
2
3
*
*
2
2
2
*
), 3
if
* ~
2
*
c +c < c +c ; 2
2
1 1
(c +c - c , min{y +y - y , 1
1
3
1 1 3
if
c +c = c +c . 1 1
Theorem kernels
23.2.
Let
* ( x ) , * (x) of (L), respect
and 9TI c
G-transforms the
form
ively,
i. e.
(Gf)(x)
(23.1)
exist
and on
2
(Gg)(x)
the
2
with
the
1
spaces
JIf (L) ci,yi
2'^2
(23.8)
2sgn(c +c ) + sgn(y +y ) > 0, k
Then
the
G-convolut
k
ion
k
k = 1,2.
k
(23.3) exists
for
all
f(x) e 3J1"1 c
€ W * c ,y 2 2
(L) if
and only
if
200
i
(L), g(x) l
(1
2sgn(c +c -c ) + sgn(y + y -y ) ^ 0, & k
(23.9)
k
3
k
k
3
1,2;
sgn(c +c -c ) + sgn(c +c -c ) + sgn(c +c +c +c ) 1
1
°
3 +
+ 2 s g n ( y +y 1 2
In
k
this
case
x
1/2
7
2
2
to
3
1
2
1
2
*+ * ~
7f ~V ) - 0.
1 2
3
-1
(f*g)(x) € !JJ1
c , y 3' °3
(L), where
(c , y ) is defined 3
by
3
^
(23.7). Moreoi/er, if x1/2(f*g)(x) € ffl-1 (L) for aii f(x) e ffl"1 (L) and g(x) G iJJf1 (L), then V*2
Finally,
there
G -transform
JUT1 (L) D JJJf1 (L). c '* V*3
is
the
(G f)(x) with
(23.3) by the product
following
the kernel
factorization
^ (T) for
of two G-transforms
property
the convolution
of
(f*g)(x)
(G f)(x) and (Gg)(x): 1
2
(G (f*g))(x) = (G f)(x)(G g)(x),
(23.10)
3
where the left
1
2
part of (23.10) is understood
(3)
by Definition
(3)
Theorem 23.3. Let the parameters a , /3 j (23.2). Let also the following inequalities i hold
,
20.2.
2sgn(c -c ) + sgn(y -y -1) > 0,
satisfy
the
conditions
k = 1,2;
2sgn(c +c ) + sgn(y +y -1) > 0, (23.11)
{sgn(c -c ♦
*
*
*
*
#
*
*
) + sgn(c -c ), sgn(c -c ) + sgn(c +c ) ° 1 3 2 3 1 3 1 2 sgn(c -c ) + sgn(c *+c*)i + sgn(y*+y*-y 3 -2) > 0. 2
Then the G-convolution
3
(23.3) can be represented
2
\R+ as
follows
201
by double
integral
on
(f*g)(x)
(23.12)
|_u vj
where G [x,y] is G-function
f(u)g(v)dudv uv
of two variables
of type (13.1]
* (s) * (t) 1 2 -s -t , ,. —-—? —^— x yy dsdt * (s+t)
G [x,y] = 2
(2iri)
3
(23.13)
p -n ^ 3
°
Q
M
0(3))
q -m ;m , n ;m ,n
3, 3
3
,
p
3
*3
l
1
2
;p ,q ;p ,q F M M l
l '*2
2
.0(3)) m +1 , q 3 3
x,y
2
(3)
i
(a
{3)
,t (a
^
)n
3
+1,p
:(a(1));(a(2))l 1 ,m 3
3
^
p 1 ^ q 1
) 1,
ro O n : 3
p ; ro \ (/3 q 2 {2)
(1)
);
Further we consider some typical examples of G-convolution and
their
factorization
G-transforms.
properties
(23.10)
with
the
)
(23.3)
corresponding
In fact the examples of §21 can also be related to G-
convolution. In accordance with Definition 20.2 we can affect by some G-transform on the convolution (23.3). Hence if G-transform
(Gf)(x) has
the kernel ^(x) (17.2) then from (23.3) we obtain the next equality
(23.14)
* (s) * (t) *(s+t) \T (_ 2 ^ — f*(s)g*(t)x"s-tdsdt. * (s+t)
1
(G(f*g))(x) =
{2niV
3
cr cr t s
It is not difficult to see that the last double Mellin-Barnes integral is a new convolution with the following factorization property
:(G
oG3)(f*g))(x)
(Gif)(x)(G2g)(x)
where the composition of G-transform in the left part of this equality is (by §18) also G-transform with kernel ^ ( T ) / # ( T ) . sum-convolution
(21.9)
can
be
obtained
by
the
For example, the
action
of
transform
(Kf)(x) on modified Laplace convolution (19.17). Thus this approach is the way of getting the new convolutions. 202
The most important particular case of G-function of two variables is the Kampe de Feriet function (1.7), which can be represented by (14.26) as follows
A:B;B' f (a):(b);(b'); (23.15)
-x,-y C:D;D'
(c):(d);(d');
(c),(d),(d')'
=r
(a)-s-t,(b)-s,(b*)-t,s,t
1
(a),(b)>(b>)
(2ni)2
r , I
x y dsdt (c)-s-t,(d)-s,(d*)-t
t
s l-(a): l-(b); l-(b')
(c),(d),(d')
0,A: 1,B ; 1,B*
(a),(b),(b')
A,C:B,D+1;B',D'+1
=r
x,y
Example r
23.1. Let
in
l-(c):0,l-(d);0, l-(d')
(23.15) x ,J y > 0 ,
L. = o* , L t
t
s
=
s
and
we
consider the G-convolution (23.3) with the next kernels ^ ( T ) , k = 1,2,3 k
B T,(b)-T
(23.16)
*
(T)
1
r(T)"TT r(b j -x)
U T~7 r(d U j -x)
=r (d)-x
B' "T^b'i-T
(23.17)
vl/
2
(T) U J
=r
= (d')-x
T1T(C -T)
(C)-T
(23.18)
J=l
* (T) 3
D'
(a)-x T(a - T ) j=l
203
j
B,
(rn q
= D+l,
a
q
= 1-b , j
j
2, .
= D'+l,
q
j = 1,
.,B';
,D+1;
(2)
0
in
Further
n
B' , n
*i
(1)
= A,
1-c
with
(23.4)
3
(OA)/2 and y , k
>
j = 1,...,B; j
1-a
, j-i
j = 1,...,A).
c* = — + (B-D)/2,
1,2,3, y
P.
/3 U ) = 1-d
,D'+1;
1
(B'-D')/2, c
B>
*2
j = 1,
j
j = 2,
accordance
= B, p
1-b.,
1
a(
1-d;^.
= C, p
3
2
c* = — +
2
2
2
are defined as in Theorems
k
23.1-23.3. Hence we can establish the corresponding results for the con volution defined below in (23.20). The factorization property and real form (23.12) for
(23.10)
this G-convolution are great interest.
In accordance with Definition 17.1
ci-(b))1>B (G
afHx)
= G
B;D + I
*• f
(23.19)
ff(u)j(x),
(i (d))
-
1>D+1
(l-(b'))
(G 2 f)(x) - G J ; J ; : + 1
1, B
1, (l-(d'))
o|"f(u)l(x)
,D'+1'
(l-(c)) (G
3f)(x)
= G
?!A
1,c 1, (l-(a))
|~f(u)~|(x)
1 , A
and by the conditions of Theorems 23.2, 23.3 the equality (23.10) holds true and G-convolution with Kampe de Feriet function
(1.7) takes the
next form of real representation (23.12)
(a),(b),(b')l r ' (23.20)
(f*g)(x) = T
(c),(d),(d')J J »
A:B;B' '
(a):(b);(b');
C:D;D'
(c):(d);(d');
F
dudv x f(u)g(v)
204
X
u'
. y V
Finally, we use the Table 4. 1 of the integral representations of Horn's
list
functions
for
the
construction
of
the
series
of
G-
convolutions in form (23.12) and their factorization properties of type (23.10). The corresponding G-transforms can be written according to the Tables
17.1
and
18.2. We
suppose
that
conditions
of
the
Theorems
23.1—23.3 hold valid and we give 11 out of 15 examples of convolutions with Horn's list functions in kernels since the corresponding 11 double Mellin-Barnes variables
integrals
(23.13).
have
the
structure
of
G-functions
of
two
The other 4 examples of Horn's functions will be
considered in §25.
Example 23.2. Let f(x),g(x) e JJl"1 (L). Then the next table of real c, d
representations (23.12) with convergence conditions (23.11) and factori zation properties (23.10), with existence conditions (17.5) of Gtransforms (G f)(x), k = 1,2 in 9J1 (L) for G-convolutions with Horn's k
c, d
list functions in kernels takes place. We also assume that the para meters of this functions satisfy the conditions
(23.2) of the poles
separation supposition (in each line of the table).
TABLE 23.1
00
" a,/3,/T" i.
00
*
r] F
(f*g)(x) = r
*
v
,
x
x
r oo> - r ^ r Adudv y;- u -,- v -^ x)f(u)g(v) l (<x,£,£ '' uv
(x1-aicx a r x r - i ) ( ( f # g ) ( x ) ) = jr(/3)(i+x)"/3|(f(x))jrO')(i+x)"/3,|(g(x )).
205
00
2.
(f«g)(x) °
= T
,
CO
F (a,|3,0;y.y' ; - - , - -
2
U V
L * > * JJ J 0
)f(u)g(v)^^-, UV
0
(( xx W = ({r[ W )) (( (( ff .. gg )) (( x x )) )) = r [ *J ]] ii F F ii o;r;-x)}(f(x)) (p:y;-x)}(f(x)) x{r[;:]iFiO';y»;-x)}(g(x)).
00 00
3.
(f*g)(x)
CO 00
= T r
F 3 ( a , a ; ^/ 3, ^^ |;y^;;-- - , - -
L y
)f(u)g(v)-^— ,
JJ J 0
0
)
A _' K x *t f( (i fg* lg K ) ) (xx)) ( xx"-*V L L 1 (l-a-|3)/2 (2X)"1 .. a e V = f| r[[^0'. aJj x Wqi -af pl =
2
^a-g '
,lA(f-r o (y-/) (J f( f( (xx )) )
2
L r , ,1 a-aij3'>/2 (2x) _1 W , 0 , , , (i)l(g(x)). x -j i op , a x e l - a - p a-/30 x J
CO 0 0
4.
( ff**gg) ()x( )x )== rr
I" a,/3
CO
1r r
[ *,?'JJ J 0
(x-(a+/3)/2LK
f_2_||
11
4
V
11V
0
x
, (l ( 1 -2r)/2 -2T)/2 f . = (x j J
xx {{
F ( aa ,, 0p ,t *r ,, *v '' ;; -- -- ,, -- -- ) f ( u ) g ( v P ^ , F
JJ yrl _i a
((aa++/3)/ 0 ) / 22) ( ( f # g ) (x)
}
,_ ,_ ^ , 1 ( y - i )) // 22 ww ,, .. ,, . w d( i--, y* ,' ) / 2 (2VxH x )(f(x))(x
(( 22 y^ x) )} lxx( (^7 -, -11) ,/ /22))(( gg (( xx )) )) ,, Reiv') [[Reir), R e ( y ) , Re(y')
206
Reij+v') ^> 1, R e(r+r')
> 3].
00
00
[r aa,/3l . p l r rr 5.
(f*g)(x) = T r
$ (a,3,y;- - , - -
& &
LL
0
U U
1
yy JJ JJ JJ 0 0
V V
)f(u)g(vP^ & &
,
UV uv
0
( x i - a i a«-yxy-i)((f.g)(x) -yxy-i)((f#g)(x)) = | r ( / 3 ) ( 1 + x ) ^ 3| |(f(x))(A ( f (x) ) ( A + g) (x) . (xi-« I ) = |ro)(i+x)-' + g)(x).
00
6.
( f *« g« ) ( x ) = r |
^
00
[ I* [* (3 3 ' y 2
L y* y
'
- - V
)f(u)H « (( vv ) ^ ^ S
U V '
JJ J 0
0
y
g ) x( x)))) ( xx-- A W_Hx (Hf (. fg» M
00
7 7..
U
{ rr (O/ 3) )( (l l++xx )) - p/ 3}| ( f ( xx )))){W r ( )^()l( +l +x x) )- -p/ '3 }' l( (gg((xx )) ) .
=
00
I" 3 1 f f (f*g)(x)
= r
L y JJ J 0
<S> ( 3 , y ; - - ,, -- -- ) f ( u ) g ( v p H ™ l , 33 U u V v UV uv
0 p = j{ rr O ( P) )( (l l+ +xx) )- - p}}( (ff((xx))))((AA+t g ) ( x ) ,
( xx -VV)y (>(
[Re(y) > 3 / 2 ] .
8.
,P
((f.g)(x) f . g ) ( x ) -^ r [f ""M lf f
d [ * v ^ ^ , , ^^. ; ;-- * *-,-*- , - * - )nu)g(v) ) f ( u ) g ( v ) d -^ - ^ ,, 1
L r.r'JJ J 0
0
(xVx«)((f.g)(x)) = {r[^]iF i O;r ; -x)}(f(x))(x (1 - 3r ' ,/2 (xVx«)((f.g)(x)) = {r[^]iF i O;r ; -x)}(f(x))(x (1 - 3r ' ,/2 \
y'-i
J
\
y'-i
J
207
9.
> x x ._, . , .dudv *T ((a,y,y ;- -,- - )f(u)g(v)
(f«g)(x) = r
2
y,y
U
(x" a A _ 1 x a )((f*g)(x)) = ( x
(1 y)/2
"
V
UV
{j y . 1 C2^)}x (y " 1)/2 )(f(x ))
(x (1 -^ )/2 {j r ,_ i (2^)}x ( ^- 1)/2 )(g(x)] [Re(y), /te(y') £ 1, /teCy+y') > 3 ] .
a, a \ fil
10.
r
(f*gHx) = r
* JJ0 y
J c
(x~*A x ) ( ( f * g ) ( x ) ) ==
x - f > r> x ,_, . , .dudv i (a,a ^,3r;- -,- - )f(u)g(v) l u v uv
{( r [ i ^ , ^ ] e - W ; ^ (I,}(f(x )) 2
jr(a , )(l+x)" a '|(g(x)).
a,/: ii
(f»g)(x) = r
" r o
X
x ._, . , .dudv
■=. (a,^,y;- -,- - )f(u)g(v)
y
2
U
V
UV
( x " V r H ( f g ) ( x ) ) = | r [ i ^ , i ^ ] e ( 2 x ) W ^ _ a (I)|(f(x))
x (A g)(x),
208
[tfe(y)>5/2],
§24. Convolutions for some classical integral transforms
As we noted above (see introduction to Chapter IV) there are some known integral transforms except Laplace, Fourier and Mel1 in transforms which
have
the
corresponding
convolutions.
These
convolutions
were
obtained in various ways and were investigated by many mathematicians.
In this section we will consider the convolutions for some classical particular
cases
of
Riemann-Liouville
G-transforms
integro-differential
transform,
the
modified
transform,
the
modified
corresponding
(see
Stieltjes sin-
and
Table
17.1)
operators, transform,
such
that
the modified the
cosine- Fourier
modified
the
Hankel Meijer
transforms.
The
results are based on the theorems in § 2 2 — § 2 3 .
But
except our approach to convolution constructions in the form of double Mellin-Barnes integrals we will use the classical way of finding the corresponding
formula
for
the
product
of
kernels
of
the
integral
transforms.
1. Case of Riemann-Liouville integro-differential operators.
Now
we
consider
the
modified Riemann-Liouville integro-differen
tial operator which is defined by Table 17.1 (line 6) in the form of G-transform (17.1) a (x' I
+
x
) (f (x) ) s G. .
|f(u)ll(x)
1,1
(24.1) ,a-8-i
|3 f (x-u) a _ / 3 _ 1 -a_ r . . =x Jo r(a-3) u f(u)duHence in accordance with Definitions 17.1, 23.1 and Mellin corresponr(l-a - T ) dence 9 from Table 18.2 we set ^ (x) = =r^—^ r, k r11-p -T J k
209
k = 1,2,3, where
(24.2) (24.2)
[ (
Re(a Re(a ) < 1/2, 1/2,
\J
Re(a )) << 0, 0, Re(a
[[
Re(0 ReO 3)) < 0.
kk == 1,2; 1,2;
Then the the G-convolution G-convolution (23.3) (23.3) of two two functions f f(x), ( x ) , g(x) g(x) e e ffl' Ml11 (L) (L) for for our case takes the form
i(a,/3)
r r rd-i53-s-t) ru-03-s-t) ni-^-s) ra-a2-t)
(f ( f g)(x) gHX) =
u^i? J J ra-a^-s-t) r(i^-s) rd-^-t) i^77
**
cr cr X. t
(24.3)
Ss
x f*(s)g*(t)x"s'tdsdt.
Since by the formulas (23.4),
K
= K y
y 3
= Re(a-fi),
(23.5) c* = 0 , / K
K
k = 1,2,3, K
from Theorems Theorems 23.1 23.1 - 23.3 23.3 we obtain the the following following correspon-
3
ding result.
Theorem 24.1. The convoluoion 3JT1 (L) c,fr
(24.3) exisss
for
f(x), g(x) e
if
(24.4)
{
2 s g n ( c ) + sgn(y+tfe(a - 0 - a +0 )) > 0,
\J
2 s g n ( c ) + sgn(2y+tfe(a +a - 0 - 0 )) > 0,
I
2 s g n ( c ) + s g n ( 2 y + R e ( a +a - 0 - 0 - a +0 )) > 0.
v v
1
2
1 ' 2
3
k = 1,2;
3
(a,13) In this +0 ) ,
cc*
Let Let (x
2
I
x
(f*g)(x)
y+tfe(a - 0 - a +0 ) ,
*J O
ft fR
case
1/2
tt t
2 t2
C t.
O O
O O
Riemann-LioaviiJe K ie^ann-Liouviiie 22
__CCXX
x
22
)(g(x)) exist
€ 3JI"1 * ( L ) , where c,y
j
= min{y+/te(a - 0 - a 1i
2y+/te(a +a - 0 - 0 - a +6 ) ) . 1
& &
1
ttrraannssffoorrmm on the
spaces 210
^
O O
((24.1) 24.1) -1 -1
O O
__
3
-.
((xx * V I1 1
!D1 (L), c,9T c,y
1i
i.e.
')f (f '\ x ^ H C x(x)) ))
and
(24.5)
2sgn(c) + sgn(v+Re(*
-$ )) £ 0, k
Then the convolution and only
(24.3) exists
for
k = 1,2.
k
all
f(x), g(x) € 3Jf
(L) if
if
[
2sgn(c) + sgn(3r+/te(ak-£k-a3+/3 )) £ 0,
k = 1,2;
(24.6) 2 s g n ( c ) + s g n ( 2 ? + K e ( a +a -/3 -/3 - a +/3 ) ) £ 0 .
Moreover,
there
is
/3
a -/3
(x
3
I
3
the following
3
property
T(a,0)
-a x
factorization
3
)((f*g)(x))
(24.7) £ a -/3 - a |3 a - 0 - a = (x *I * *x 1 ) ( f ( x ) ) ( x 2 I 2 2 x 2 ) ( g ( x ) ) . Finally,
if
the following
inequalities
hold
ReU -0 -a +0 ) -1 > 0,
k = 1,2;
(24.8) Reio. +a -0 -0 ) -1 > max{0, Re (a -0 ) +1}. 1 2
Then
the
convolution
3
1 2
(24.3) can be represented
3
by double
integral
on
2
IR+ as
follows
(24.9)
where
«£<*> = {} G[n # ^ > ^
G [x,y] is the next
(24.10)
G-function
of two
0,1;0,1;0,1 G [x,y] = G 1, 1; 1,1; 1,1 x,y
variables
0 , 0 :a ;a 3
1 2
a :0 ;0 3
1
2
More interesting situation is represented in the case with the same kernels iels for convolution (24.3) when a = a = a = a, 0 1
2
3
Besides the Theorem 24.1 we will use Theorem 22.2. 211
= 0
= 0
= 0.
Corollary 24.1. Let Re(oc), Re{fS) < 0; a = ^ 0^ = QL^ Re(ct), Re{(3) ag ==a 30^; ; 00==01^ = 0 =fi. The convolution 2
(24.3) exists
for f ( x ) , g(x) e € 9JT1 (L),
3
if if
c,3f
sgn(3-+tfe(a-0)) ^ 0, ( 2sgn(c) + sgnij+Re(*-$)) \ [ 2sgn(c) ++ sgn(2*+fle(a-0)) sgn(2*+fle(a-0)) ** 0. 0. [ 2sgn(c)
((24.11) 24.11)
jCa.0) ^a.0) and x 11 // 22 (f*g)(x) (f*g)(x) 6 5J1"11 and e 5JT
„ R
L)).. ((L
miinn cc ,, m 2 ^ + R e (( a a -- 0 0 )) >
Moreover, there
is the following
facoorization
propprty
jCa.0) (x/3Ia"/3x-a)((f*g)(x)) = (x/3la-/3x-a)(f(x))(x^la-/3x-a)(g(x)).
(24.12)
+
+
+
Proof. From the Theorem 24.1 and the Definition
15.2 it is not
difficult to show that the conditions (24.4) are equivalent to (24.11). Hence the condition a
(24.5) for the Riemann-Liouville transform
(24.1)
a
(x^I ~^x~ ) of the functions f(x) and g(x) holds valid and we obtain the corresponding factorization property (24.12). Corollary 24.1 is proved. ■
The real representation (24.9) does not exist in this case since the conditions (24.8) are false but in accordance with Theorem 22.2 we can establish
the corresponding
representation
(22.13)
for convolution
(24.3) in case of the same parameters. In our case the Theorem 22.2 is reduced to the following form.
Theorem 24.2. Let conditions
(24.11) hold
f(x), g(x) € UK" UJT1 (L), Jte(a), Re(oc), /te(0) Re(0) < 0 and valid.
Moreover,
if
the next
inequalyty
the takes
ace
frS J.
(24.13) then the next j = 1,2,3) 1,2,3J) is
2sgn(c- 1/2) + sgn(y//te(|3)) ^ 0, representation
of the convoluoion
true 212
(24.3) (a. = a, 0. = 0,
r~ w ^ r a /3-a - 0 . , 2a-2 A 2-2a. (( f * g ) (x) = (x TI' x ' )o(x Ax )c
u
(1-u)
(24.14) , 0-1.-1 l - 0 _ w . , 3-lA-l 1-3 W ,„ * A l x (y1 A y 'f)(xu) (y' A y ' g) (x(l-u) )du
where
the
operators
in
the
right
part
of
(24.14) are
defined
in
§17 — § 1 8 .
Similarly,we can obtain the corresponding constructions for adjacent Riemann-Liouville
operator
(x I
x
)(f(x))
(see
line
8
from
Table
18.2). We note the simplest case of convolution for this operator if in kernel
*(T) =
o+ ) we
r
set
a = 0,
0 = 1.
Hence
from
(22.1)
and
Definition 15.2 the above-mentioned convolution can be represented as follows l i
(24.15)
(f*g)(x) = (lV 1 )(f(x))g(x) + (lV 1 )(g(x))f(x)
and its factorization equality takes the next form 1
(24.16)
1
l i r
(I x" )((f*g)(x)) = (I1x"1)(f(x))(I1x"1)(g(x):
2. Case of the modified Meijer transforms.
From the Table 17.1 (line 23) we have the modified Meijer transform as follows
(24
17)
J2Kv (2Vx)lo[f(u) I )I
= G'2,0 0,2
>|"f(u)j(x i^/2,-i^/2
00
=2
R
(2vx7u)f(u ,du
In formula (23.1) setting * (x) = r(-iV2+T)r(iV2+x) for all k = 1,2,3 we establish the next convolution for modified Meijer transform
213
r(^/2+s)r(-v/2+s) r(iV2+t)r(-iV2+t)
1
(f*g)(x) =
r(i>/2+s+t)r(-i>/2+s+t)
Uni)' (24.18)
f*(s)g*(t)x"s Theorem 2 4 . 3 . Let for
all
\Re(v)\
f ( x ) , g ( x ) e ffi"
1
c, y
< 1. Then
the
and only
if
convolution
(24.18)
exists
2 s g n ( c ) + s g n ( y - l ) £ 0.
(24.19)
In this
(L) if
\isdt.
case
x 1 / 2 ( f » g ) (x) e UTT1
(L).
c.min (y+l , 2^+1 )
For convolution
(24.20)
(24.18) the factorization
property
holds
J2K^(2v^)jo["(f*g)(u)l = J2K^(2v^)U Tf (u)l J2K^(2v^)|o ^ ( u ) ! .
Proof. This theorem is the corollary of the Theorem 22.1, where the conditions (22.4) are reduced to inequality (24.19) since K = 1, JI = 0, V* = - 1 .
-
For convolution (24.18) the corresponding real representation (22.8) does not exists since here conditions (22.7) are
false
and
integral
(22.9) diverges. But we can obtain the analog of the Theorem 22.3.
Theorem 24.4. Let
\Re{v)\
< 1 and f(x), g(x) € 3Jf
(L). Then
the
c, W
representation
of
the convolut
ion
(24.18) is
true l l
(f*g)(x) =
X I
X
|o|x
I
X
(u /u ) 1
2
(24.21) du du
V/2
(1-u )/(l-u ; 1
2
flA u u |g lGl(l-u 1 2J
K
214
1
1
2
)(l-u )I u u (1-u )(l-u ) • 2
J
1 2
1
2
next
3. Case of the modified Stieltjes transform.
The modified Stieltjes transform can be found
in the Table 17.1
(line 21)
l.lf
^ l
(l+x)"Pj<>rf(u)l = G***
ir{p)
(24.22)
|f(u)l(x)
0
u P f(u) du
r(p)
(x + u) p
U
It is more convenient to denote the transform (24.22) by (S f)(x). Hence P setting in (23.1) * (T) = r(x)r(a -x), k = 1,2,3 we establish the k
k
convolution for modified Stieltjes transform in the next form
r(s)r(a -s) r(t)r(a -t) (24.23) (f*g)(x)
-
f (s)g (t)x
cisdt.
r(s+t)r(a -s-t)
UniV
3 t s
In accordance with (23.4)-(23.5) we have for convolution (24.23) that c r = 1-Re(a l-Re(oc c* = 1, y* k
k
) , k = 1, 1,2,3 and y = y . From Theorems 23.1 - 23.3 ), kk
3
3
we get the following result. Theorem 24.5. Let Re(oc ) > 1/2, j = 1,2, Re(oc ) > 1. The tion
(24.23) exists
for
all
1
f(x), g(x) € JJT (L) if
and only
convolu
if
c,y
2sgn(c) + sgn(r+/te(a - a )) £ 0,
k = 1,2;
(24.24) 2sgn(c) + sgn(2r+l+ite(a - a - a )) £ 0.
In
this
( f * g ) ( x ) e 3JT (L), c,r# j + Reioc -a ), 2y+l+Re(a -a -a )}. 3
case
2
Moreover, (24.25)
x
3
there
is
* *
= min{y+Re{oc - a ), 3 l
1 2
the following
factorization
s a (S (f»g)(x)) = (S f(x))(S 3
where
1
g(x)). 2
215
property
Finally,
if
the
following
inequalities
hold
Re(a -a ) > 1, 3
(24.26) Reia Then
the
convolution
(23.12) on IR+ with
-a -a
3
l
2
) > 1.
(24.23) can be corresponding
k = 1,2;
k
represented
G-function
of
by
double
two variables
integral (23.13).
Now we consider the convolution (24.23) for a 3 = a +a 1 2. It is not difficult to see that the last inequality of (24.26) is false in this case. Therefore, the real representation (23.12) on (R+ does not exists. Nevertheless, we have the corresponding relation as in (22.20).
Theorem 24.6. Let € TF1 (L). Then c,r true
(24.27)
the
Reict
) > 1/2, j = 1,2, a j
next
representation
of
a -l a -l ,/ u u 1 (1-u ) 2 f
(f*g)(x)
the
convolution
, 1-u
>.
This formula
can be
x
obtained
(22.10), Definition 15.2 and Fubini theorem.
j
\
by
and f(x), g(x)
1 2
(24.23)
^
i A A-T=£-
\
Proof.
= a +a 3
\
using
is
du du
x
(1-u2
u >
\
the
i
beta-integral
■
If, instead of convolution (24.23), we consider the next integral
(T(s)r(a -s) r(t)r(a -t) (24.28) (f*g)(x) = (2ni ]
f*(s)g*(t)x"s_tdsdt, r(-l/2+s+t)r(a +l/2-s-t) 3
(r cr t s
where Reict
) > 1/2, then we can also obtain the theorem of existence and 3
factorization property in form
(24.29)
x" 1 / 2 (S
a
(y1/2(f*g)(y)))(x) = (S
a f)(x)(S a g)(x). °
°
3
1
216
2
Moreover, we can evaluate the corresponding G-function of two variables (23.13) by means of double hypergeometric type series (see (8.18)).
Let
the
kernel
of
the
convolution
(24.28) be
#(s,t)
and
the
corresponding G-function of two variables G [x,y] be defined by means of formula
(23.13). The integral
Re{f3),
Reir)}
(23.13) for G
[x,y] converges when min
> 1/2, x,y € IR+ and the conditions
(24.26) are
true.
The purpose of this point is to find the representation of G [x,y] by means of Horn's list of functions
(see §4). The method which we use
contains the repeated applications of the Slater theorem
(0.I.Marichev
(1983)) for getting the value of Mellin-Barnes integral from the sum of the residues in the left (or right) poles. For this we will subdivide a quarter of the plane R
into four domains: { 0 < x , y < l > ,
{ 0 < x <
1 <
y>, {0 < y < 1 < x >,{1 < x, 1 < y}.
1. Case
0 < x, y < 1.
As Re{oc ), Re{oc ) > 1/2 then the integral of 1
2
the type (23.13) with the kernel & (s,t) can be evaluated repeatedly from the sum of the residues in the left poles of the gamma-functions at first for s = -m and then t = -n. Finally, we get the next representa tion
*'*.*-£ ^
A" r a
a +m,
a +n
1
2
+ — +m+n, - — -m-n 3
2
2
(24.30) a ,a 1 27T
2.
1
,a + — L
where F
3
3
1
F ( — , a , a ; — + a ; x , yJ ) ,
2
2
1
2
2
3
2
is an Appell series (A.Erdelyi et al. (1953)).
Case
0 < x < 1 < y.
In
this
two-dimensional
Mellin-Barnes
integral of the type (23.13) can be evaluated from the sum of residues in the left poles s = -m and the right poles t = a +n and we will have
217
i
/-r
\
G [ x , yJ ] = )
(-1 J
m+n
. a +n m,1. 2
l_^ m!n!
a +m, a +n, 1
_
—;—:— x ( - )
r
y
2 l
l
a + — +m-a - n , - — -m+a +n 3
2
2
2
2
m , n=0
(24.31) a , a
=r
1
1
2
,
3
2
2
1
3
G (a , a , — + a - a , — - a 2
— +a - a , a - — 2
where G
2^
yJ
1
1
2
2
2
3
2
1 . y
; - x , - — J,
2
2
is a Horn series (A.Erdelyi et al. (1953)).
3. Case 0 < y < 1 < x.
Here the next result is true
a ,a G [x,y] = r
1
1
2
1
— +a - a , a - — 2
3
1
1
2
(24.32)
x
G (a , a , — + a - a , — - a 2
4. Case
1 < x,
1 < y.
1 2 2
=
;- —,-y). 1
X
The following result takes place
a G [x,y]
1 3 2
-•
,a 1
2K1
2
?+-
(24.33) -oc
-<x 1
xx
2
y
. „ r- (
3
.
a
.
,
1
^
.,
1
^
F ( — +cx +a , a , a ; — +a +a +a ; — , - ) . 1
1 2
2
1
2
2
1
2
3
x
y
The value of the function G[x,y] at x = 1 (or y = 1) is defined by the continuity of G[x,y] and can be obtained from Gaussian summation theorem (8.20).
For the case of the same kernels (a = a, j = 1,2,3) of convolution j
(24.23) as it is not difficult to see that conditions (22.7) are false. Hence the real representation (22.8) does not exist and we can apply the Theorem 22.3 for getting the relation (22.20).
Now we distinguish a more interesting case of convolution for
ordinary ordir
Stieltjes
transform
(24.22)
(p = 1),
when
(24.23)
a
= 1 j
j = 1,2,3.
218
Hence, using the addition formula for gamma-function we can obtain the next representation
sin(n(s+t))
sin(7is)cos(7rt )+cos(7rs)sin(7it)
r(s+t)r(l-s-t)
r(s)r(i-s)r(i/2+t)r(i/2-t)
r(t)r(i-t)r(i/2+s)ru/2-sr
Then substituting this identity in (24.23) and applying Definition 15.2, s the convolution (24.23) (f*g)(x) for a. = 1, j = 1,2,3 brings to the following one
(f*g)(x) = 7T f(x)
2712
r(t)ra-t) • . -t,. € r(i/2+t)r(i/2-t).g (t)x dt
(24.34)
r(s)rd-s) * . -s, 2rTi r(i/2+s)r(i/2-s).f (s)x ds
+ g(x)
Further, since
O.I.Marichev
r(s)rd-s) r(i/2+s)rd/2-s)
ctgdrs)
and the next
integral
(see
(1983))
s-l
(24.35)
exists
1-U
in principal
du = 7i c t g U s ) ,
value
in point
0 < fie(s) < 1,
u = 1 we
can prove
the
following
Parceval equality of type (15.4) in space !IJ1 (L)
(24.36)
2TT
i
r(s)rd-s) * , -s r(i/2+s)rd/2-s)1 lsJX as
219
=_^
n
f(t) t-x
dt = (Sf)(x), x > 0,
where (SO( x ) is
a
singular operator on half-axis (0, +00) (see S.G.Samko,
A.A.Kilbas and O.I.Marichev (1987)).
This fact
can be established
by representing
the right
part
of
equality (24.36) as follows x-l/P
f(t)
(24.37)
t-x
dt =
N
f(t)
lim
t-x
M, N, P—X» 1/M
dt.
x+l/P
Hence applying the Definition 15.2, changing the order of integration and using the uniform boundedness of inside integrals by the aid of Lebesgue theorem we establish (24.36).
Thus the convolution for Stieltjes transform (S f)(x) takes the next form
(24.38)
(f»g)( x)
= Trjf (x)(Sg)(x) + g(x)(SO(
.,].
The equality (24.36) allows to solve the question of evaluating the norm of singular integral operator(24.36) in 3R~ (L).
Actually, from (15.6) and the inequality |ctg(7rs)| < 1, s € (r we have +00
(24.39)
||Sf| _,
= ||ctg(Ti(^-+it))||f*(^r+it)dt|
im'V)
|f*(-L+it)|dt
l f l -x ' m (D
Hence |S| ^ 1. But if we consider the next sequence of functions JR (U f (x) € im'^L), n = 1,2,..., where
220
t f *(
r i . • J. \ %» +lt))
JL+it) = <| COS(TI( 2
t ^ -n,
2
t > -n, then it is not difficult to obtain that for any e > 0 there exists such number n, that
n
Thus US!
m a(D
n
1
m
(D
=1.
4.Case of the modified Hankel transform.
The modified Hankel transform can be defined by line 9 of Table 17.1 and line 13 of the Table 18.2 as follows
f (a-/3+i)/2. (n/-\\fr( ^ _ ~1,0 J |x a+P-i(2^)}(f(x)) =G0,2 (24.40)
>|f(u)l(x) [ a,l-0J
00
_
"
x
JVi (2vx/u) u
f(u)—
Then the G-convolution (23.3) of two functions f(x), g(x) € OTf (L) for our case takes the form
r rifi -s-t) r(a +s) r(a +t) (24.41)
(f*g)(x)
' 2
(2iri)
3
1
2
r(a +s+t) r o -s) r o -t)
t s
x f*(s)g*(t)x"s_tdsdt. For
this convolution all
transforms approach
of
can
easily
obtaining
results just as
be the
obtained.
Now
convolution
the Hankel transform (24.40), when a
221
in the previous we
for
= 0, 1-0
consider one
the
particular
= -v,
cases of classical case
of
j = 1,2,3. Hence,
if Re{v)
^ 0 from a stronger
O.I.Marichev
theorem
and S.B.Yakubovich
than 17.2
(see Vu Kim Tuan,
(1986)) it follows
that
the Hankel
transform can be represented in the form
(24.42)
f J / ) ( x ) = |x" y/2 J^(2v^)|(f (x ))
J (2/~^7u")u
f(u)du,
v
x > 0.
In accordance with the Theorem
17.1 if f(x), g(x) € UJ1 (L), then
|J f|(x), (j g](x) € S J T ^ ^ ^ a )
and
0, Re
(V)
by
Consequence
15.2
h(x) = x 1 / 2
(j fl(x)fj gl(x) e 3TT1 , (L). IV J { V J 0, Re (V) The inverse transform to
J f (x) denote by
to the Theorem 16.4 for f(x) € 3H
(x) and according
(L) -v,0
f (x) = G0,1 2,0
W
(24.43)
J f
)
>[f(u)j(x)
For our future convolution it is natural to require the fulfillment of
the
factorization
property
of
type
(22.6).
In accordance J
Definition 20.2 we represent the convolution
x- 1/2 [3^h)(x) c),
where
the
operator
with
v
(f*g)(x) sought for as
fj^fl(x) fo r
f(x) e
ffl
(L)
0,Re(y)
(similar to (24.42) for 0 < Reiv)
(24.44)
M"i
(x) = X
^ 1) has the form
J (2/u7x)u '*
w
"f(u)du,
x > 0.
Thus, we have the next repeated integral for convolution (f*g)(x) 222
(f*g)(x) = X
r-\(
T M/
,
.-V/2.-1/2
1/2
J (2v u/x)(x/u)
u
J (2VHLI7V)(U/V)
(24.45)
,-v/2 , ,dw " g(w) — du — . x f(v)dv JV (2vu7w)(u/w) ° w u
—1
Theorem —
24.7.
Let
f(x),
1^/p-^/A
g(x) € Jli (L)flL(x
< Re(y) 5 1 Then x1/2(f*g)(x) e !JJ1 * (L) and the following
;R ).
and
represen
tation is true
(f*g)(x)
r(,)x-2p [ fr l-(Vx/u 2TT r(2v) J J L
- Vx/v)"
D(x,u,v) (24.46) (vxTu + vxTv) -1
where the domain
(24.47)
Proof.
D(x,u,v) is defined
/x/u - vx/v
u
v
by
the
f(u)g(v)dudv,
next
x > 0,
inequality
< 1 < vx/u + vx/v
In accordance with the conditions of the theorem and
asymptotic behaviour of Bessel function it is not difficult to shown that the repeated integral (24.45) absolutely converges. Hence by Fubini theorem we change the order of integration and the inside integral is the known Sonine integral
(see formulae 2.12. and 41.14, Vol.2 from
A.P.Prudnikov et al. (1986))
223
V-l/2 V-l/2 (
0 l-3l'
r
?
2 -(a-b)22 cc2 -(a-b)
-i J
VH abc)"r(i>+l/2)L V^ ((abc)^r(i^+l/2)L 00 00
I
J
1^-1/2
r
v-l/2
V 1_y
u~ u JJ \
V V
(au)J (bu)J (cu)du == J V V
xx [(a+b)22-c22l , .
V V
if |a-b|
J
|_
a,b,c > 0, Reiv)
o 0 (
Thus by
the simple
change
0, 0,
> -1/2,
in in other other cases. cases.
of variables
in
integral
(24.45) and by
the evaluation of Sonine integral we may come to representation (24.46). j
v
The statement x1/2(f*g)(x) s 3T-1 (L) and The statement x1/2(f*g)(x) e Jj-1 (L) and Thoperty Theperty
corresponding corresponding
factorization factorization
(vi),,,. (V]<«>M<"
(24.48)
can easily be obtained by Theorem 17.1 and Consequence 15.2. Finally,
we
try
to
distribute
the
convolution
■
(24.46)
on
the
modified Sin-transform
00 (X)
G
This
n,o o[f(u)](x) = — i - sin(2v^Al)u"1/2f(u)du, 0> 0> 2[0,-1/2j 2[0,-1/2j L L -I I vv^xj ^J o
transform
can
be
obtained
from
(24.40),
(v = 1/2). But in the Theorem 24.7 Re{v)
when
x > 0.
a = 0,
0 = 3/2
> 1/2 and we will consider the J
corresponding convolution as
(f*g)(x) = lim (f*g)(x). From representa1/
tion (24.46) we get the next convolution 224
7 1 / tL +
Si n
(24.49)
-1/2
(f*g)(x)
u
2VTC x
-l/2_,
v
^
^,
f
■,
„
f(u)g(v)dudv,
~
x > 0,
D(x,u,v)
where the domain D(x,u,v) is defined by the inequality (24.47).
The convolution for cosine Fourier transform is easily obtained by simple
trigonometric
formula
for
product
of
two
cosines
(see
also
I.N.Sneddon (1951)).
§25. Modified H-convolution
The idea of this section is based on Remark 15.2 where we have noted that it can be established a more general property of belonging to space 5H
(L) of the product of two functions. Similar
to the problem of
c , <J
function product convolution
in these
object.
Hence
spaces we
can
it
difficult
is not
consider to
the
corresponding
bring
the
double
integral (20.3) to the next modification of H-convolution
& (s) $ (t) 1 2 * . * . -As-Wt, ,, , ,. _,_ . , f (s)g (t)x dsdt, $ (As+wt) °
(f*g)„ Cx) = ,_1 .,2 (A,w)
(25.1)
(2711 )
3 cr cr t s
where $ ( T ) , k = 1,2,3 are defined by (20.1) and A,w € (R, Aw * 0. This k
modification
is very useful
for obtaining
some new convolutions and
their factorization equalities which will be considered below. According to conditions
(20.2) the corresponding
inequalities for
parameters of the modified convolution (25.1) will be written as follows 225
k t( Re(^ )) k)) tfe(/3<
I I
(25.2)
1 I I
! k l) >>0, 0, ^i bb |
j = 1,2 m k, , j 1,2,...,m k
k = 1,2,3; k
k> lk) 1-Re(«! )--La! k) > 0, 0, l-Re{alk))-±-a j 2 j j j 2
j 1,2,...,n j = 1,2 n , , k k
k = 1,2; 1,2; k
++
, „ , (3). (3>A+U 3
1l _-
„.
R e t a )^_ W ' ^
Re ( a j
>>
aj - r
0,
°-
Re(a<3>) > 0, 0> Re(a'3)) + + a a !"3>) ^ ^> J J *J J *i3) (3)
( 3 )
j
j
I l-^e(/3 l-Re{(B )) - b
j j=1 1,2,.,.,n ,2 n,, , 3
j p , j = = n +1 +l,...,p 3 3
3 3
^ > 0, 0, jj = m 33 + ll
qqg3.
2
Similarly to asymptotic estimation (20.6) for the kernel in convolution (25.1) we have $ (s) $ (t) VAs^t) "Se*P
, >n [-(sIs.tl-Kjsl-^ltlJj
r
-hxi^n *Eexp Ksi-ti-Kjsi-^itiJj 3
x |s+t| 3 | s | 3
a a
x |s+t| IsI
It1
2
It1
2
, ,
where the ordered pairs U ,/i ), k = 1,2,3 are defined by (20.4), (20.5) k k and the value JI is evaluated follows where the ordered pairs U ,/i as ), k = 1,2,3 are defined by (20.4), (20.5) k k and the value JI is evaluated as follows 3
{3)
~* = ji* = M 3++^|li ^ " M^ £ 3a(3) a(3) _ - E 3 bl\( 3 )} . [j=i J j=i J J
(25.3)
Now
we formulate
the corresponding
representations for modified H-convolution
theorems
of existence and
(25.1) which take the forms
of the Theorems 20.1-20.3 in case X = w = 1 and they can be proved in the same way but only with the help of the corresponding estimations in §20.
Theorem 25.1. The convolution
g(x) € 3JT1
c ,y 2 2 2 2
(25.1) exists
(L) if
226
1 for f(x) €ffi ffi" * (L), (L), c ,y c1 .y, 1
C +K
2sgn( l^,1 - K 3 ) + sgnCy^ f^-J^) * 0,
f
C +K
2sgn( ? ,2 - K 3 ) + sgn(r2+ ^"M*) * 0, C +K
(25.4)
C +K
2sgn(-|^p + - 4 ^ 4 ) + sgn(ri+ * 2 + jn^ JLX^ ) ^ 0, C +K / 1 1
sgn(
_K )
^r
C +K , 2 2
x ,
sgn(
3
C +K r 1 1
v
C +K 2 2.
T^r " K3} + s g n ( ^q- + -ra- }
+ 2sgn(? + y + M + l^~M ) - °-
In this case x (c , y ) is defined 3
(f*g) .
as
(L), where
(x) <= 5J1
(A, w)
c ,3r 3
the
pair
^
3
follows
3 r
C +K.
(
^ (Cl+Kl)/lAl < ( C 2 + K 2 ) / I W I
-|AT" V W V '
C +K
( f ,2 3
3
K
U
(25.5) (c ,r )
3
, y +ji -M*), 2 *2
i f (c
3
+K
2
2
Theorem 25.2. Let H-transforms
lT (C +#C }/ X
1
1 I I
-l
2
$ (x),
2
$ (x)
(20.1)
(L), respectively,
3J1
)/|A| 1
l
i.e.
exist
=
1
(C +K )/ W
2 2 I I
(Hf)(x) and (Hg)(x) with 1
1
+K
l
'
'
C +K
VWVV 3 '
kernels
) / | w | < (c
on
the
the
2
spaces
the conditions
O T C
c ,y l i
(20.20) hold
(L)
and
valid.
the
Then
2
convolution
25.1) exists
for all
(L), g(x) e m
f(x) e 311 1 1
and oniy if 227
L) if 2
2
C +K
^
2sgn( K . 1
- K3) + s g n ( r a + J ^ " ^ )
C +K
2sgn(
* 0,
^#
- T 3 T " K3)
+ s
e n ( V f2-»*3) * °.
(25.6) C +K , 1 1
Sgn(
v
-Txr " K 3 )
+sgn(
,
C +K 2 2
" T ^ ~K3]
. ,
+ sgn(
,
C +K 1 1 ,
C +K 2 2.
3
nrr ra^
+ 2sgn(y i +y 2 +M 1 + M 2 -M 3 ) * 0.
/n this case x ( A + W (c ,y ) is € m'1
defined
(L) for
by
1)/2
(25.5), and
f(x) e 5JT1
all
(x) e UJf1
(f*g) .
moreover,
(L), where
3
3
if
x
the pair
(f*g)
(L) and g(x) e 3J1"1
(x)
(L), then 3H"1 (L) D
(L).
mr3
3
Finally,
there
E-transform
is
the
(H f)(x) with
the
following kernel
$ (T) for
3
(25.7)
3
the
left
Remark
15.2.
convolut
ion
of (25.1)
)(x) = (H f)(xA)(H g)(x W ),
° (A,W)
part
of
Theorem 25.3. Let conditions
the
property
3
(H (f»g),.
where
factorization
(25.2). Let
1
(25.7) is
the also
understood
parameters the
2
following
228
a. ,
by
Definition
20.2 and
/3. , a. , b. satisfy
inequalities
hold
the
2sgn(ic -1X|#c ) + sgn(ji -fi - 1 ) > 0 2sgn(u , - | W | K 3 ) + sgn(/i2-ji*-l) > 0, (25.8)
2sgn(K +K ) + sgn(ii +u -±, - 1 ) -> ^, 0 1
1 2
2
2min^sgn(K njsgn(Ki-|A|jc — |A|#c3) ) ++ sgn(K sgn(ic2 "|W|K - | w | i c3 ),) , sgnCic sgn(ic -|A|K - | A | i c ) + sgn(ic +*c ) , sgn(K ) + sgn(K +K )i + sgnt/^+^-fi*K 2-|w|ic 2 ~ I W I K 33 ) + S g n ( l C l i+ I C 22 ) | + S S n ^ 1 + ^ 2 " * 1 3 - 2 ) > ° -
Then
the
IR as
convolution
represented
by double
integral
on
follows
^,M^=ff«[^>'^
(25.9)
where
(25.1) can be
H[x,y] is H-function
of two variables
(20.27).
Now we consider some examples of modified convolution (25.1) in form (25.9) and factorization property (25.7).The corresponding conditions of validity can be obtained by the Theorems 25.1 - 25.3.
Example 25.1. Let A = w = 1/2, $ (x) = $ (x) = $ (x) = T(x). Then 1
2
3
the integral (20.27) is convergent by the conditions Legendre's (21.13)
doubling
formula
for
gamma-function
(25.8) and by
and
identity
the corresponding H-function H [x,y] is equal to H[x,y]
r(s)nt) x -s y -t d s d t
= {2niY t s
M«L I ^ t i r ^ 2 (2*i)2 I I F ( s + t ) t s
2Vii
exp
mn 229
tdsdt
Hence we obtain the convolution
eXp
= ^ _
(25.10) ( f * g ) ( 1 / 2 > 1 / 2 ) ( x )
[" X H
f (u) g (v)
dudv uv
with factorization equality
(25.11)
(A (f*g) )(x) = (A f ) ( v 5 ) ( A g ) 0 / x ) . + (1/2,1/2) + +
The more general by Gauss-Legendre
example of convolution
product
formula
(25.10) can be
for gamma-function
(see
established O.I.Marichev
(1983)).
Example 25.2. $ ( T ) = T(x). r = 2,3,...
Let
Then
and
A = u = 1/r,
the
using
the
r = 2,3
integral
(20.27)
reduction
property
H-function can be represented by the following
7—x
i2ni)
r
is
* (T) = * (T) = 1 2
convergent
(13.6)
the
for
y
dsdt
r
-1/2, _ .(l-r)/2 (27i)
r(s)f(t)
r(s+t)
UniV
^.fs+t+j)
I I1 Iv r IJ (x/r)
(y/r)
j=i
t s
=r
1/2,_
. (l-r)/2 r r - l , 0 G 0 , r-1
(2TT)
m'
Hence we obtain the convolution
230
l/r, 2/r,
all
corresponding
equalities
r(s)nt) -s -t, ..
H[x,y]
(25.12)
and
(r-l)/r
dsdt
t r-
f*g&
t
A (l/r,l/r)
^
1/2,
,(l-r)/2
0 (x) = r (27 i)
(25.13)
jr-1, 0 J 0 ,r-l
"(j ^ ) r
1/r, 2/r,,
(r-l)/r
dudv f(u)g(v) uv
with factorization equality
(25.14)
)(x) = (A f)(rVx)(A g)( r Vx).
(A (f*g) +
( l/r, l / r )
+
+
It is necessary to note that the case r = 1 brings to the kernel of modified Laplace convolution (19.16), but the corresponding H-integral (20.27) diverges due to conditions
(25.8). Therefore
the convolution
(19.16) has no representation (25.9).
Example 25.3. Let A = w = — , $ (x) = $ (T) = T ( T ) and $ (T) = r
2
1
2
3
r(x)/r(i;-T). Then similarly we have the next formulas
(f*g) &
(x) (1/2,1/2)
(25.15)
hffl ]
T(y+l/2)
2Vn~
UK
J 1>-1
r-f , f ,1/u+l/v, , f (u) to g (v) dudv, uv
(2rfol x(y"1)/2)((f*g) to J
(1/2,1/2)
( x ))
(25.16) = (A f ) ( V x ) ( A
g)(i/x).
Now we return to §23,
where it is noted that in Table 4.1 there are
four examples of Horn's functions, which have the integral representa tions not satisfying the structure of G-convolution 231
(23.3) (see lines
5 — 6,
14 — 15).
But it is not difficult
to find that two functions
H (a,/3;y,«5;-x,-y) (line 6) and H (a; y, 8; -x, -y) 4
(line 15) satisfy
the
7
structure of modified H-convolution (25.1) when A = 2, w = 1.
Example
25.4. The
next
pairs
of
convolutions
and
factorization
equalities
a,0
{f
*z\2
(25.17)
„
(x) = r 1}
f
s
0
x
*
\r-r
\
r
^dudv
// (a,£;y,5;- -,- - )f(u)g(v) 4
(x"aA'1x<X)((f*g)/
U
&
V
UV
(x))
(2,1)
= {rg] i F i O;y;-x)}(f(x 2 )) (x ( 1 " 5 ) / 2 { J ^ t t V x ) } x (6 " 1)/2 ) (g(x)),
(f*g) (2)i) (x) = r
(25.18)
r ,8
„ , ~ x x »_, . , .dudv // (a;y,<5;- -,- - )f(u)g(v) , 7 u v uv
Re(y+<5) > 3,
(x~V 1 x a )((f*g) /
(x))
(2,1)
(x<1 r)/2
" { v/2^}*'9"1'^'^^5
* (x U - 5,/2 { J a _ i (2^)}x <3 " 1)/2 )(g(x)),
take place for some functions f(x), g(x) € !Jfi
(L), where the parameters
c, d
of space
and
of
the kernels
satisfy
25.1 — 2 5 . 3 .
232
the conditions
of
the Theorem
§26. General Leibniz rules and their integral analogs
In
this
last
section
we
consider
the
original
application
of
G-convolution (23.3) to constructing new Leibniz rules for some Mellin type transforms, namely G-transforms (17.1).
Leibniz rules for the operators of fractional calculus and their integral analogs were considered by several authors (see, for example, Y. Watanabe (1931), T.J. Osier (1971), (1972a,b), S.G.Samko,
A.A.Kilbas
and O.I.Marichev
17.1
(1987)).
operators are known
In fact, as is shown in Table
to belong
these
to the class of G-transforms. Conse
quently, it seems very natural to try to obtain some new Leibniz rules and their integral analogs for the others G-transforms similarly as for fractional integro-differential operators, i.e. as series or integral representation of some G-transform of the product of functions f(x)g(x). At first in papers of S.B.Yakubovich and Yu.F.Luchko
(1991a,b) it has
been found a general method for constructing such Leibniz rules and their integral analogs on the basis of notion of the G-
convolution
(23.3). We will use the following formulas and summation theorems in the further discussion
(see also L.J.Slater
(1966), A.P.Prudnikov
(1986), Vol.2.):
(26.1)
T(z)rd-z) = 7T/sin(7rz)
(Addition formula for gamma-function).
Gaussian summation theorem: a,b; (26.2)
F 2 1
1
[c,c-a-b] [c-a,c-bj '
233
Ke(c-a-b) > 0.
et al.
Saalschutz*s summation theorem:
(c-a) (c-b)
-n,a,b; (26.3)
F
n
1
3 2
n
(c) (c-a-b)
c,1+a+b-n-c;
Dixon*s summation theorem:
a, b, c; (26.4)
[l+a/2,1+a-b,1+a-c,l+a/2-b-c] [l+a,l+a/2-b,l+a/2-c,1+a-b-cJ'
F 3 2
1+a-b,1+a-c;
fle(a-2b-2c)
> -2.
Dougal1's formula:
(26.5)
1 f r[ a + n ^ + n l = . . f. . h , r['[c+d-a-b-1 c-a,d-a,c-b,d-b /
|_c+n,d+nj
sin(7iajsin(7ibj
[_<
Re(a+d-c-d) < - 1 ; a,b £
(26.6)
|_a+u,b+u,c-u,d-uj
[a+c-1,a+d-1,b+c-1,b+d-lj'
fie(a+b+c+d) > 3.
c+d-2
(26.7)
(26.8)
c+u,d
] du = —T(c+d-l]
[~a+iu,b+iu,a-iu,b-iu~|
L
_
J
Re(c+d) > 1.
_
i-2a-2b 3/2 f"2a,2b,a+b] n
Reia) > 0, Reib) > 0. 234
[ a + b + 1 / 2 J'
We also introduce the following operators, which are the particular cases of G-transform (17.1) in addition to Table 17.1:
1) The operator with Tricomi function ^(a,b,x) in the kernel and its inverse: , a.b -a_. , . -2,0 (x * x f )(x) = G. ' a 1,2
fa+1+a-bl
Jf(u)j(x
-x/u_, . . . -a-i _, x , e $(a,b,x/u)u f(u)du,
( a,a+l-b * l\ '
2) The
(x
operator
Al x v
x"af)(x) =G1-°\ '
with
f)(x)
algebraic function in the kernel:
= x
M
(1+•!-x/u)
+(1-Vl-x/u)
u
f (u )du
A/1-x/u -v/uL
2Vn
3) The
o f(u) (x); L -I
a+l+a-b
operator
with
_ , a+i>/2,a+iV2+ • 20 , 0 | 2 2,2
G!
generalized
Laguerre
|"f(u)j(x);
polynomial
L (z) k
kernel:
(x L x k
f)(x)
1
G
r
l,l
fa+1,a+A+1
Tf(u)l(x)
TT 2,l
e
a+k+A+1
-u/xT A., . v -a-i_, . , L (u/x)u f(u)du; k
235
in
the
4) The inverse operator to the operator with Kummer function
F (a;b;-z)
in the kernel: , a( T(b) r b V1 - a _ w , t-TUTFa } X f H x )
(X
=G
_0,1 2,1
f oc+l-b,a
>|~f(u)l(x) a+l-a
5) The operator with Gauss hypergeometric function
F (a,b;c;x) in the
kernel: (a+l-a.a+l-b) a,b
r(a)
2,2
>|f(u)l(x) a,a+l-c
[/.
(a,b;c;-)H(l- -) + x =-, TFTTT—£T u u r(c-a)r(l+a-b)
F (a,l+a-c;l+a-b;-)H(- -1 )]u"a_1f(u)du. 2 1 X U J
Then
setting
in (23.3) ^ (s) = $ (s) = 1 and using
(26.1) - (26.8) for representations
the formulas
of the G-convolutions
kernels in
(23.3) we obtain the following theorems.
Theorem 26.1. Let f € m
(L), g € 3R
c ,7 1 1
(r for (f*g)(x) (23.3). Then the following definitions
of the operators
1. If H(x) = r(y-ct-0-T),
(L) and *
c ,y 2 2
Leibniz
3
rules
(x = H x , X € hold true
also see in Table 17.1J:
then
(xa+/3 *A x r " a ^)(f(x)g(x)) = (f*g)(x) (26.9) ^ ^
7-1 o 7-1 o , a-7. 7-a _ w w £-7. 7-$ w . (x 'L^ x° f)(x)(x' °L k x° 'g)(x)
236
(the
under
conditions: Re{?-ct-p)
> 1,
Reioc) > -— ,
Ke(|3) > - — ,
(26.10) 2sgn( C i ) + sgnCr^eCr^jpJO+l) * 0, i = J U .
2. If
H(T) = r(r+S-a-/3-l-x), (x
' A x
then
^
) ( f ( x ) g ( x ) ) = (f*g)(x)
(26.11) 1 = V (xy"V+^"axa-yf)(x)(x5"V+a-^-5 /^ r(l-a-n)r(l-^-n)lX Vp X *Mxnx *n+a x
under
*)(x) g^xj
conditions: Re(y+8-oc-p) > ±- , a,0 * Z, /te(y-a) > - ^ - , /te(r~0) > - y-
(26.12)
Jte(S-a) > - — , Re{6-&) > -— , 2sgn(c i ) + sgn(y i + Re(a+/3-2J y }o) £ 0, i = J 2 | -
3. If H ( T ) = l / T ( r - 0 - x ) ,
then
(x y ^ 1 x ^ y ) ( f ( x ) g ( x ) ) = (f*g)(x) (26.13)
= l
r J
^ ^
a
- r
}x-af)(x)(x^{Fy.a}x^g)(x)
k=0
under
conditions: Reiv-p)
(26.14)
> - \ .
^e(a) > - ■!■ ,
Ke(/3) > — , 2sgn(ci-{1f})
fle(y-a-/3)
+
> -—,
sgn(V*e{y«a}
237
+
{_? } ) , 0. i = { > } .
4. If H(x) = r(a+/3+n+x)/T(a+/3+x),
then
(-i)"(x-^mnx-^),f.x)g(x)) (26.15)
= (f*g)(x)
n
= (-l)n ye(x a (-^)Va)(f(x))(x ptn (-^) n "V P )(g(x)), = ( - 1 ) nkV= 0 C^(xa(4j)kx"a)(f(x)) (x'3+n(^Lpx-'3) (g(x)), k=0
where
k
C are binomial
coefficients
n
under
n € N,
fte(a)
fie(a+/3)
> - — , 2 s g n ( c . ) + s g n ( y . - n ) * 0, i = 1,2.
(26.16)
> -— ,
Re((S) Re(j3)
conditions:
22
> -— —, 22
(.1)^ ^ -- (( A A || nn (( xx -- ^^ ff (( XX )) gg (( xx )) ))
= (f*g)(x) (f*g)(x)
(26.17)
k=0
(
~
} X
~dx
' f(x)g(x))
=
(f*g)(x)
n
(26.18)
= = («): (aj^^f-DviyV/jyifix))) x ^(-D e;(y ( x- (y (f(x))) 1
a+P+n+1
k
n
k
a
k
k = 00
x
under
(^_] k (x a+n- 1 f^l n (x- a -^ (x )))
conditions: ReW Reifi)
<< --—\ ,,
Re(a+0) > - — , ReU+P)
2 s g n ( c . ) + s g n ( yy . - nn)) £ * 0,
(26.19) i = 1,2, n e I N ,
a * 0, Re(oc) Re(a) >
238
i
i
-—. 2
5.
If
H(x) = r(l-a-/3-T)/r(l-/3-x),
then
(x^" a x a,3 )(f(x)g(x)) = (f*g)(x) (26.20) 00
P k a a p
- I (:)(■*)<•<(x))(x I - x - )(f(x)). 5
k=0
where
are generalized
ReU-fS)
binomial
coefficients
under
conditions:
> 0, Re{(3) < 0,
(26.21)
+ sgn(
^^-{wzY
V Re{ S }} * °- i = {2}'
6. If H(x) = r(l-a-^+y/2+T)/r(l+y-a-^+x), then
(x1+9r/2-a-|3I3r/2xa+/3^-1)(f(x)g(x)) = (f*g)(x)
(rd + r/2)) 3 00
(26.22)
x )
' ' t r ^ ) ( y / 2 + k)2
7
r/2,l-a
k!
i.y/2
«- T / 2 -x
l+y/2+k,l-y/2-k
k=0
, y/2+1-£ l+y/2 ^-y/2-1 , , > (x F x g) (x) l+3r/2+k,l-3r/2-k °
under
conditions:
Reiy/2-oc-p)
>--?-,
Ke(y-a-0) > - - ? - ,
Re-j^}- > \- ,
(26.23) 2 s g n ( c - — ) + s g n ( * . - 1+ Re{y)/2 i
2
239
+ KejjH) - °» i = - u } -
7. If H(x) = r ( r + 5 - a - 0 - l - T ) / r ( S - 0 - T ) , (x' (26.24)
I
a
xa
then
) ( f ( x ) g ( x ) ) = (f*g)(x)
+oo
V^ fy-a-ll , i-5Tn+a 5 - a - i w _ , . w £ ..1-7-11 y - £ - i w , x. = > , (x I+ x ) ( f ( x ) ) ( x ' I+ x° ' ) ( g ( x ) ) + n=-oo
under
conditions: Re(y+S-a-/3) > 2, Ke(S-/3) > 1, Re(5-a) > — , 2
(26.25) ite(a) > 0,
2sgn(Ci - \ )
a * IN, fle(/3)+ —
* IN,
Re{y-{3) > — , * Re(fS) > - — , fle(S) < — , Re(y)
< 2,
+ sgn(yi+ *efcjl + 1) * 0,
i = jU-
8. If H(x) = l/(r(a+/3+T)r(l-a-0-x)), then (xa+P|(l+x)'1|x"a"P)(f(x)g(x)) = (f*g)(x)
(26.26)
= J(x a {(l + x)-j~x-^ +
under (26.27)
(x 1/2+a {(l + x)-j"x- 1/2 - a )(f(x))(x^(l + x)^
conditions: -—
< Re{a+p) < 0,
2sgn(c -1) + sgn(y ) £ 0, i = 1,2.
9. If H(T) = l/(r(S-a+T)r(y-0-T)), then
(x5'a|r(5+r-a-^)(l+x)a+P"5"y| xa"5)(f(x)g(x)) = (f*g)(x)
240
]
= rty^-a-g-D 2 ^ ( x |n8-g+l)(l+x)
(26.28)
jx
(f(x))
n = -oo
x
under
(x1-a{r(y-«+l)(l+x)a-y"irIxa+n-1)(g(x))
conditions:
(26.29)
fie(a)- — <£ I, 2 Re(8-oc) > - — ,
Re{(3)+— £ Z, ' 2 Re(?-/3) > 1, /te(5-|3) > 0,
fte(a+0-y-5) < - 1 , 2sgn(c.-l) +
Remark (26.20),
26.1.
We
note
that
■*><*,♦ ^ a } ' ^ °- i = f2}-
formulas
(26.24) can be easily reduced
for fractional
(26.15),
shall
demonstrate
the
(26.17),
(26.18),
to known classical Leibniz
integro-differential operators
Proof. We
Re(y-a) > 0,
typical
rules
(see references above).
moments
these relations on the examples of the formulas
in
the proofs
(26.20) and
others are proved in the same manner. At first we will prove
Let us take a = -a, b = t, c = 1-/3-S in the formula
of
(26.24).The (26.20).
(26.2). Then we
obtain the following representation:
r(i+a-/3-s-t) _ rq+q-ft-s) ra-/3-s-t) ni-0-s) 2i
-a, t; l-/3-s;
(26.30) r(l+a-/3-s) r(i-|3-s)
I k=0
Since the gamma-ratio ^ of #
this
theorem
then
(-a) (t) k
k
Re(oc-fS)
> 0.
k! (1-/3-S) k
(s+t) = H ( s + t ) , where H(s) is defined by case 5
the
G-convolution
(t) = 1, takes the next form
241
of
type
(23.3),
where
# (s) =
(26.31)
P
(f*g)(x) = = — —wl+o P ?/) f-t) f*(s)g*(t)x-s f*(s)g*(t)x-s Sisdt. \isdt. (f*g)(x) T[!+o 1 _ P3 3 (27ii)2J J r' (27ii)2J J r ( 1-/ 's "-^ t) (T (T t s
It follows from Theorem 23.1 that G-convolution conditions
(26.21) and the double
integral
(26.31) exists under
is equal
to the repeated
integral. Hence similarly (20.13) we have the representation
T ) ( ) x T ddT, xl/2(f*g)(x) = U J nr }!; gn- T^ ; 7F FTT)*~ xl/2(f*g)(x) = —^2712 " T> rci/2-^-x) (
P
(26.32)
(T (T
where
TT
F(x) = -- U U f*(x-tt— )g*(t)dt
(26.33)
27li
T
for
T T
2 t
Further, from the Theorem 15.3, the Mellin correspondence 9 from the Table
18.2 and
the Definition 20.2
it
is not difficult
to get
the
following equality
(f*g)(x) = (x/3IVxa_/3((f(x)g(x)).
(26.34)
++
Now using
the representation
(26.30) and
substituting
the series in
(26.31) we have:
(f*g)(x) (26.35)
+oo
k =. _- Ji _ [f [ rE f - o g - c u dxsdt. -w "( 1i;T; *y:y^ fY (-a)k k(t)k f*(s)g*(t)x— (27ii)2 J J J J (2Tti)2
r(1
cr
We
will
show
that
"^-s) "P-s)
one
integration and summation
can
LLk = 0
k=0
k!(l-p-s) k-d-p-s) k
change
k
the
order
in the right part of
of
operations
of
(26.35) by means of
Lebesgue theorem. At first, we estimate the partial sum S (s,t) of the series 242 242
s(s,t) = p T ^ y—-—^f*(s)g*(t)x-s-t . 1Z (-a) (t)
n
(26.36)
S(s,t) = r T ^ 7 T 1U li SJ
^f*(s)g*(t Ix" 5 ^
"
I— k!(1-0-s) k=0
k
Re is) = — , Reit) Re{t) = — , x > 0. It is convenient to represent when Reis) S (s,t) in the form: N
>t) SN( - r(i-p-s) N( ss,t) rd-p-s)
s
L ,(i-p-s) 2- kk-(i-p-s) k =0
k
Jt (-a) (t)
(26.37)
+
s ^ ^?Z?^ 7£h^ ± fi * t)x ",x~ " t m f (' s( , 8g ,( g (t " 't
k=N
1
'
k
= S (s,t) + S'(S,t), S'(s,t), N
N
1
where N is a constant, which we will define in the further discussion. At first, we estimate every term s (s,t) of the finite sum S k
(s,t),
N
1 using tne ioiiowing simple consequences o± tne biining s iormuia. and using tne ioiiowing simple consequences o± tne btining s iormuia ana
the Definition 15.2: Re
(( 22 66 -. 33 88 ))
| \ v^\ g Wj C| s \
(26.39)
k k |™*>| |r[^k)[ = |t| |t| [i|\ ♦+o0d(/l |/t| |t )| )] j, ,
(26.40] (26.40)
II ((-_(X a )) I
k!
I k!
I I
k - s -s_t t
x"
|
(26.41)
*
y y nc I s II
* f
(s)|s|
l
e
1
a f ( s ) | s | *2*e"c t g (t)|t| 2e 2
g (t)|t|
e
'
i fi f I t| tI |—> —>0000, , t t €€(T(T. .
i f Re(s) = _L , Reit) i f Reis) = — 2 , Reit)
< C,
I
if IS I ^ —> « 00 ,s , S G ,.. (T . if |s| 6 1
=
F ( s ) 6 L(cr ) , s
= F ( s ) e L(cr ) , = G ( t ) € L(
= — , x > 0. = — 2 , x > 0.
s
Then we obtain, that |s (s,t)| < C |F(s)G(t)| and
(26.42)
IS 1
N
(s,t)| < C |F(s)G(t)| e L(tr x (r ). '
1
3'
'
t
s
For estimation S'(s,t) we will use the following simple inequalities:
(26.43)
|T(x) I * TiReM)
(26.44)
|B(x,y)|
if Reix)
,
r(x)r(y)
> 0.
< B(/te(x),/te(y))
f(x+y)
if Reix)
> 0,
Reiy)
> 0.
Then we have in particular:
(26.45)
|r(l+a-/3-s)r(-a+k] T(l-/3+k-s)
r(l/2+/te(a)-/te(f3))r(-/te(a)+k) r(l/2-/?e(|3)+k) if k > Reoc,
(26.46)
|r(t+k)I * r( — + k ) ,
k = 0, 1,2,
k = 0,1,2,..,
| g (t)/r(t)| ^ c G (t)
(26.47)
where G (t) e L{cr )
If we choose N true for any k £ N
(see the conditions (26.21)).
= max{l, [/te(a)]+l>, then inequality (26.45) holds and using (26.43) —
(26.47) with conditions (26.21),
the next estimations take place
|s;(s,t)| =
rd-/3-s)
( a)
k(Uk
Y—-
r(l+a-0-s) Vx
l—
* . * . . -s-t -f (s)g (t)x /3-s)
k!(l-
* If (s)|| g (t)/r(t) x
f(-a)
244
r( —+/te(a)-Re(|3))
V^ r(-/te(ct)+k) r (1/2+k) l^ r(l/2-Re(p)+k ) k!
(26.48)
I
s Cs|F(s)Gi(t)|
-Rect+Refi-l
> k
C | F ( s ) G ( t ) | e L(cr x
1
'
t
s
Finally, in view of (26.37), (26.42), (26.48),we can apply the Lebesgue theorem and rewrite (26.35) in the form:
(f*g)(x)
r(l+a-|3-s)(
I—
a)
kU)k
*
r(i-p-s) k!d-p-s) f
. * .. - s -t. ,.
(s)g (t)x
dsdt
t— (2wi) cr cr t s
(26.49)
where
the
operators
I
(-a)
£
[ I )(x*"k Ik"a x a - /3 )(f(x))(x k (^) k )(g(x )),
k!
last
k
1
r(l+a-/3-s) _ _ ^ _ _ _ f *( s ),x -s, ds x
2ni
equality
as G-transforms
1
(t)
g (t)x" clt
2TTI
follows in
the
from
the definitions
Table 17.1. Comparing
of
fractional (26.34) and
(26.49) we conclude, that Leibniz rule (26.20) holds true under condi tions (26.21).
We will prove the formula (26.24) now.
Let us take a = a, b = /3+t, c = r, d = 5-s in the formula (26.5). Then we obtain the following representation:
245
r(y+5-l-a-/3-s-t) _ sin(7ra)sin(7cQ+t))r(y-a)r(a-a-s)r(y-/3-t) r(5-|3-s-t) (26.50) y /
x
The corresponding
G-convolution
r a+n,/3+t+n 1 |_ y+n, 5-s+n J
(23.3)
^ ( ^ 5 . ^ ) > 2.
(see case 7 of
this
theorem)
takes the form
(26.51)
(f*g)(x) =
r(5-g-s-t)—
(27ii)2
f (s)g (t)x
dsdt
-
cr cr
t s
It follows from Theorem 23.1 that G-convolution
(26.51) exists under
conditions (26.25) and we can represent it in the next form
(26.52)
(f.g)(x) =
(x0-S+V
+a
- V r + a - a - * - 2 )(f(x)g(x)).
Substituting the representation (26.50) in (26.51) we have
(f*g)(x) =
sin(7ia)sin(rc(f3+t) )r(y-a)r(5-a-s)r(y-/3-t)
1 {ZniV t s
(26.53)
\^ _f a+n,£+t+n 1 *, , *,., -s-t, ,, ) T ' f(s)g(t)x dsdt. & / [ y+n,5-s+n J
We
will
show
that
one
integration and summation
can
change
the
order
in the right part of
of
operations
of
(26.53) by means of
Lebesgue theorem. For the sake of brevity we introduce the following notation:
246
,
h
t)
n
sin(Tra)sin(Tr(/3+t))r(y-a)r(5-g-s)r(y-/3-t) 2
I" a+n,/3+t+n 1 [ 7f+n, 5-s+n J
x f^slg^Dx"5"1.
At first, we estimate the partial sum S (s,t): N
n
(26.54)
S N (s,t) =
) hk (s,t),
Re{s)
if
L.
= 2— , fie(t)
, x > 0.
It is convenient to represent S (s,t) in the form: N
N
-1
(26.55)
S (s,t) N
= £\(s,t) + £\ s,t) + ) h
k
k=-N
/ *—' k=0
1
k
s,t
= S
N 1
s,t
+ S'(s,t). N
We estimate the sum S'(s,t), using the following simple relations in N
accordance with conditions (26.25):
(26.56)
r(y-(x)sin(7ra)
-s-t X
2
if Re{s)
l'
Re(t)
— , x > 0
A
2
(26.57)
r(a+k)r(<5-a-s) T(5-s+k)
is a constant. 1
r(ftea+k)r(Re(6-a)-l/2) r(tfeS-l/2+k)
if Re (a) > 0,
Re[8-a)
> — , k
0,1,2,
2
(26.58)
|TO+t+k)| * r(ReO)+-i-+k),
(26.59)
|f (s)| < F 2 (s),
if Re(fB)
where F (s) e L(cr 2
247
s
> - —
(26.60)
|r(r-0-t)sin(ir(0+t))g (t)| * A G (t),
where G ^ t ) e LCo-J.
Then, we obtain that
|S'(s,t)|
V h (s,t) * ) h (s,t)
1
L
N
'
■I
(26.61)
k
\ L\ k
r(y-a)sin(7ra)
r(r+k)
-s-t
r(a+k)r(5-a-s] |r(/3+t+k)| r(5-s+k)
If (s)| |r(y-0-t)sin(nO+t))g (t)|
/2+k) V" r(Jte(<x)+k)r(/te(g) + l * A F (s)G (t) l^ r(Re(5)-l/2+k)r(r (y+k) 3 2 2
But the last series in (26.61) converges under conditions (26.25) since
k*%'vRe(oc+(S-8-y) '" ~ B'+1fl + 0(l/k)
r(/te(a)+k)r(Ke(l3) + l/2+k) = = r(fie(S)-l/2+k)r(y+k)
Thus
S'(s.t) 1
N
< A '
F (s)G (t) 4' 2
2
For estimating the sum
]•
€ L(
t
s
S (s,t) we represent it in the form: N 1
5 (s t) = V 5
'
N
sin(7ry)sin(7r(5-s))r(?-a)r(5-a-s)r(?-/3-t
/
2
(26.62) _f l-y+k,l-S+s+k "I *
. *
r[ x-^.x-p-t+k Jf (s)g Similarly,
we
use
the
corresponding
.-
(t)x
relations
for
gamma-functions
and functions f (s), g (t), which are provided by conditions (26.25). Hence, it is not difficult to establish the next estimation 248
|S ( s , t ) | * A 5 | g ( t ) | | r ( 5 - a - s ) s i n ( w ( 5 - s ) ) f
(s)|
1
V"
(26.63)
r(l-Re{y)+k)r{3/2-R -/te(S)+k) a+k)
l^
r(l/2-/?e(|3)+k)r(l
k=l oo
< A F (s)G ( t ) 6 3
)
n
€ L(cr x cr ) .
3
Thus, u s i n g L e b e s g u e theorem we can r e w r i t e
) (y+n)
1 2711
n=-oo
2711
(26.53)
in the
form:
r(S-a-s) * , -s, =7^ -—c— f ( s ) x d s f(5-s+n)
r(y-/3-t)
* .,
-t,.
ra-p-t-n)
g (t)x
dt
(26.64)
E(^n:i)(xl"5"ni"ax5"a"1)(f
(x))
, (B+n T l-9T-n r-fB-1 w , x x X (x' I X ) (g(x)) .
Comparing
(26.64) and
(26.52) we conclude, that Leibniz rule
(26.24)
holds true under the conditions (26.25).
Similarly,
using
the
definitions
1) — 5 )
of
G-transforms
and
summation theorems (26.2)—(26.5) we can establish the other Leibniz rules in cases 1 — 4 , 6, 8, 9 of the Theorem 26.1, namely:
1) applying addition formula for gamma-function (26.1) we can obtain the rule (26.26);
2) in Gaussian summation theorem
(26.2) setting a = a+s, b = t+/3,
c = y we can obtain the rule (26.9); setting a = a+s, b = /3-t, c = s+y we
249
can obtain the rule (26.13); setting a = -n, b = -t-0, c = s+a, we can obtain the rule
(26.17); setting a = -n, b = t+0, c = 1-n-s-a we can
obtain the rule (26.15); 3) in Saalschiitz's summation theorem (26.3) setting a = s, b = -t-0, c = s+a we can obtain the rule (26.18); 4)
in
c = -t+0
Dixon's
summation
theorem
(26.4)
setting
a = y,b = -s+a,
we can obtain the rule (26.22);
5) in Dougall's formula (26.5) setting a = a, b = f, we
can obtain
the
rule
(26.11);
setting
d = 6+s we can obtain the rule (26.28).
Remark
c = y+s, d = 5+t b = 0+s,
a = a-t,
c = y-t,
■
26.2. The others series representations
of
G-convolution
(23.3) can be received, for example, from the values of hypergeometric functions in particular points (see A.P.Prudnikov et al. (1989)). The formulas (26.6) —
(26.8) can be applied for obtaining the integral
analogs of Leibniz rules for some G-transforms.
Theorem 26.2. Suppoee = HH(x). ( T ) . Then 1 1. If
the
following
f e Wf 3H"11
that integral
analogs
(L), g € 3 3JT J1 *1 of
Leibniz
(L) and
rul2s
H(Th = the(a+0-l+x) the ( CL+B— 1 + T ) tien tien H(Th
( xa+/aR+ / V x (x V x (26.65)
R '
'
)(f(x)g(x)) = (f*g)(x) )(f(x)g(x)) = (f*g)(x)
+ 00
_ 22-a_/3
(xa+TA _1x"^"Tf) ( 2x) (xa" T A _1 x~^+T
)(2 Id
-00
under (26.66)
conddiions: Re(a+0) > 1,
2sgn(c -1) + sgn(y ) £ * 0, i = 1,2. i
250
i
hold
* - 1 (x) true:
2 . If
H(x) = r ( a + | 3 + y + 5 + l - T ) / r ( | 3 + 5 + l - T )
then
, -/3-5 T -a-3r a+/3+3r+5 w _ r . , . . _ . (x I x ) ( f ( x ) g ( x ) ) = r( f * gw) ( x ) (26.67) Ia+3^| * -^-x T T-3r £ + 3 ^ , ^ , -<5+TT-T-a a+5 w . , I (x I °x' ° f ) ( x ) ( x I x g)(x)dx
under
conditions: Re{oc+(3+y+8) > 0,
(26.68)
fie(a+S)
Re{(3+8) > 0,
> - —,
ReO+y) > - — ,
iiB+ ]
2 s g n ( c - — ) + s g n ( * - Re\ 1 2
3 . If
i
H ( T ) = r(a+/3+r+S-l+x)
, a+^+y+6-i.
(x (26.69)
Ax
*
0,
i
(a+s: then
-a-^-y-5+i
_. . , . .
w
,_
.
) ( f ( x ) g ( x ) ) = ( f * gw) ( x )
+ oo
—,
1
ww^
, oc+tTy-8+i - a - 3 r _ w
r(x
r(^-x)r(5-T)
under
Re(a+y) > - 1 ,
.
X
T-6+1
w
'f)(x)(x'
fi+8r8-y+i *
x-^+i
-(B-8
X
conditions: linx. min^Re(a+y) , R e ( a + 5 ) ,Re(/3+*) , Re(/3+S) V > - -
( 2 6 . 7 0 ) Re(a+fi+y+8)
> — , 2 s g n ( c )+ s g n ( * - Re 2
i
i
2f
251
(2a+*+S) (2/3+r+6)
) *
w
._.
g)(x)dT
°
0,
4. If
then
H(x) = l/(r(a+5+l-x)rO+y-l+x))
(x
(26.71)
^ + 3r-i| r(a+ ^ + ^ +g)(1+x) -a-^-2r-6| x - / 3 - y + l ) ( f ( x ) g ( x ) )
=
(fWg) (x)
13 5 1 _ r(/3+5)r(a+y) (x/3+T|r(/3+5+l)(l+x)-|3-5-1U^-T)(f(x)) r(a+/3+y+S-l)
^jro+s+Dd+x)- - - }"^-
(x*"T{r
x (x<,"^r(a+y+l)(l+x)"a"9r"1i xT_3r)(g(x))dT
under
conditions: Re(a+/3+?+S) > 1,
( 2 6 . 72)
/te(|3+y) >
min^ nirw.Re ( a + 5 ) , R e ( a + * ) , Re(|3+5) 2sgn(c.-l)
+ s g n ( * . ) £ 0,
5. i f H ( T ) = r(a+£+x)/r(a+/3+ — +T)
—,
}>o,
i = 1,2.
then
, a+/3Ti/2 -a-0-i/2Wx., > , x, (x I x )(f(x)g(x), (26.73) , a-ixA1 (x Al 271
under
x
-a+ix _ w w 0-iXA. f)(x)(x' Al
2iX
-0+iX w » , x' g)(x)dx 2iX
conditions: Re {a) > - — ,
Re(a+/3) > - — , 2
(26.74)
2sgn(c -1) + sgn(? -1) ^ 0, i
Re({3) >
2
i = 1,2.
i
Proof. For example, we consider
the proof of rule
(26.65).
The
others are proved in the same manner. Let us take c = s+a, d = t+/3 in the formula (26.7). Then we obtain the following integral representation: 252
(26.75)
1 = 22-oc-B-s-t T(a+£-l+s+t]
dx r(s+a+x)r(t+/3-x) '
Reiot+p)
> 0.
The corresponding G-convolution takes the form
(26.76)
1
(f*g)(x)
r(«+g-l+5+t) f Cs)g (t)x-s"dsdt.
(2iri)'
It follows from Theorem 23.1 that G-convolution
(26.76) exists under
conditions (26.66) and by equality (18.3) we can represent it as follows
(26.77)
(f«g)(x) = (xa+/3 1A lx
a /3+1
)(f(x)g(x)).
Now we use the integral (26.7) for the kernel of G-convolution
(26.76)
and obtain: * . * . -s-t 02-a-/3-s-t 2 f (s)g (t)x
(f*g)(x) (2712)'
(26.78) &z dsdt. r(s+a+x)r(t+/3-x]
We
will
show
that
one
can
change
the
order
of
operations
of
integration in the right part of (26.78) by means of Fubini theorem. For the sake of brevity we introduce the following notation:
(26.79)
F(s,t,x,x) =
2
f (s) g (t) x r(s+a+x)r(t+/3-x)
We consider the following integral now
(26.80)
I(x) =
F(s,t,x,x)dxdsdt. (T (T -00
t s
253
It i s convenient to represent
I(x)
i n t h e form
F(s,t,x,x)dTdsdt +
I(x) =
F(s,t,x,x)dxdsdt cr (T - N t s
(26.81)
F(s,t,x,x)dxdsdt
= I (x) + I (x) + I ( x ) , 1
2
3
CT (T N
t s
where N is a constant, which we will define in the further discussion. For estimating the integral I (x) we have |l 2 (x)|
|F(s,t,x,x)Idxdsdt +
|F(s,t,x,x)Idxdsdt
|F(s,t,x,x)Idxdsdt +
|F(s,t,x,x)Idxdsdt
v
(26.82)
v >r -N
I
(x) + I
21
>r (x) + 1
22
where s = — + iu,
(x) + I (x), 23
t = —+iv,
2
v >r -N
24
r is a fixed constant.
2
Hence we must show that I2j (x) < +oo, j = 1,2,3,4. This fact follows from the conditions
(26.66) and the next estimations of integrand in
each integral (26.82):
for I (x): 21 2-a-/3-s-t
0 2
'
*
.
*
r(s+a+x)r(t+/3-x) where B
.
-s-t
f (s) g (t) x
< B If*(s)g* (t)| 6 L(cr xcr x(-N,N)), 1'
is a constant,lul < r,IvI < r;
254
'
s
t
for
I
(x):
2-a-/3-s-t * . * . -s-t 20 f (s) g ( t ) x r(s+a+T)r(t+£-T) where B
for
is a constant,
I
|u|
> r,
^ _ If
(s) g ( t ) I
Hs+a+r)
v
< r;
3i
. ,
,
M KI.
.
1e L(VV(-N,N)).
(x): 23
2
' f (s) g ( t ) x T(s+a+T)r(t+^-T)
where B
f
rit!AS t ) l
*UVV(-N.N)).
is a constant, |u| < r, |v| > r ;
for I (x) 24
* . * . -s-t f (s)g (t) x r(s+a+x)r(t+/3-T)
02-a-/3-s-t
2
where B
< B
f * (s)g* (t) r(s+a+x)r(t+/3-x)
€ L(or xcr x(-N,N)), s
t
is a constant, |u| > r, |v| > r.
Then we will estimate I (x). l
(26.83)
|F(s,t,T,x)IdTdsdt
I (x) l
|F(s,t,-T,x)Idxdsdt. cr cr N
(T (T -co t s
t s
Hence, using formula (26.1) we obtain the following representation:
|F(s,t,-x,x)
22"a"/3r(l-s-(x-HT)sin(7i(l-s-a+x))f*(s)g*(t)(2x)"s"t T(t+/3+x)
Thus, using inequality (26.44) we get 255
22"a"/3r(l-s-a+T)sin(7r(l-s-a+T))f*(s)g*(t)(2x)"s"t r(t+0+x)
*
^ T(l/2-Re(a)+T)- . , ,„ , * , , ■ g (t) i r(l/2^e(3) + T)l S i n ( 7 r ( 1 - S -" + T ) ) f (S) I r(t+/3+A)
C
where A is a constant and A > - - - Re(6), 2
- -fte(a)+x > 0. Hence, 2
|F(s,t,-T,x)|
g (t) f(t+/3+A)
€ L(cr x
t
if we choose N = max{l, Re(oc)-—
, - — -Re(ft)}. 2
can be obtain
in the same
The estimation of I (x)
2
manner
3
as I (x), if N = max{l, - R e ( a ) - — , 1
-_L
+
2
Re(/3)>.
Thus
if we
- — +/te(/3)>
choose
Re{oc)-
N = max{l,
— -/te(0),
-Re(a)- — ,
2
then all integrals in the right parts of (26.82),
2
(26.83)
converge and using Fubini theorem we can rewrite (26.78) in the form
(26.84)
^2-a-0
(f*g)(x)
(x
Comparing
similarly
A
x
other by
integral using
g (t)(2x)'
ru+p-x)
A
x '
that Leibniz
dtdx
g)(2x).
rule
(26.65)
(26.66).
analogs
the
1 27ri
f)(2x)(x'
(26.77) and (26.84) we conclude,
holds true under conditions
The
f (s)(2x)" s ds r(s+a+r)
1 2?ii
in this
formulas
definitions of G-transforms, namely: 256
theorem
can be
(26.6)—(26.8)
and
established corresponding
1) in integral (26.6) setting a = a+1, b = /3-s+l, c = y+1, d = 5-t + l we can obtain the analog (26.67); setting a = a+l+s, b = /3+1+t, c = y, d = 5 we can obtain
the analog
(26.69);
setting a = a+l-t, b = £+s,
c = y+t, d = 6+1-s we can obtain the analog (26.71);
2) in integral
(26.8) setting a = s+a, b = t+/3 we can obtain the
analog (26.73). Finally
we will
formulate
algorithm
for
the
evaluation
of
some
classes of integrals and series with respect to the indices (parameters) of
hypergeometric
functions
as
the
application
of
the
Theorems
26.1 — 2 6 . 2 .
ALGORITHM FOR THE EVALUATION OF INDEX INTEGRALS AND SERIES
First we derive some corollaries from the Theorems 2 6 . 1 — 2 6 . 2 . If functions f(x) and g(x) in these theorems are Meijer G-functions (1.3) then we have the following statements:
1) one can verify the conditions in these theorems without any difficul ties;
2) the result of applying G-transform (17.1) to Meijer G-function is the new G-function;
3) G-convolutions
(23.3) of Meijer G-functions are G-functions of two
variables (13.1).
For example, Leibniz type rule (26.9) will have the following form
257
I
1+/3-?, (c ) , 0
1+a—y,(a ) , a , r + l , l + l J p u+2,v+l a+k,(b )
1 _m+l,n+l r ( y + k ) k !.G p+2,q+2
u
0+k, (d )
q
(26.85)
l+a+0-y:(a );
0, l : m , n ; r , 1
= G
p
1, 0 : p , q ; u , v crx, wx
: (b ) ; q
(c )) u
(d ) v
under these conditions: Reiv-oi-fi)
> 1,
2sgn(c.) + sgn(y.-l)
Re {a)
Re((3)
> -■
> 0,
>--!-,
i = 1,2,
{;}»-»• >-©•
2sgn(c.) + sgn(y.-Re(y-2-
where pairs (c ,^ ) are defined by (17.4) i
i
The same corollaries have been obtained
for others Leibniz type
rules and their integral analogs.
Then the algorithm for the evaluation of index integrals and series is divided into four steps:
1. Representation of functions in the integral or sum in terms of Meijer G-functions.
2. Search for the appropriate Leibniz type rule of the form such as (26.85).
3. Representation of index integral or sum in terms of G-function of two variables.
4. Representation of G-function of two variables by means of more usual special functions.
We illustrate this algorithm by the following example: one needs to evaluate the given sum: 258
k!
q = V
p(y-i,a-a-k)f
^r(^+k) k where P
I
)
2
!+ bx J
(y-i,c-|3-k)( k
I
_ 2 _ | + dx
1
J'
(z) is the Jakobi polynomials (A.Erdelyi et al. (1953)).
1. We use the following representation in terms of G-function
(y-i,a-a-k)( k [
2 1 + bxJ
_ (Ubx) a t Y ' a (bx)" a 2,l bx r(a+y-cc) k! 2,2
l+a-y,a a+k, a
under conditions: Reia)
Re(y-a) > — ,
> -— ,
/te(c) > --
2
/te(a) > - — ,
/te(0) > - —
2
2. One can see that this sum may be evaluated by means of the Leibniz rule (26.85) under condition
Re(y-a-/3) > 1.
3. We
the
obtain
the value
of
sum
in
terms
of
G-function
of
two
variables:
S =
(l + bx) a + *" a ( l + d x ) c + y - * T(a+y-a) r(c+y-/3)(bx)a(dx)°
l+CL+(S-y:
^0,1:1,0;1,0 1,0:0,1;0,1
bx.dx : a ; c
4. We represent
the resulting G-function
of
two variables using the
property (13.7) in terms of more simple function:
S =
(l+bx) a+ *
a
(l+dx) C+3r ^ r(y+a+c-arCa+y-a) r(c+y-/3) (1+bx+dx)3r+a+c-a-£'
259
Bibliography
Agarwal, R.P.(1965). An extension of Meijer's G-function. Proc. Inst.
Sci.
India
Part
Aizenberg,L.A. and Yuzakov, A.P.(1979). Integral residues
in
Nat.
A 31, 536-546.
multidimensional
complex
representations
analysis.
Nauka.
and
Novosibirsk.
(Russian).
Anguino
M.E.F.
de.(1975).
On
an
integral
transform
kernel of Mellin-Barnes type integral. Kyungpook
Math.
J.
involving
15, 175-181.
Appell, P.and Kampe de Feriet, J.(1926). Fonctions queset
Hyperspheriques;
Polynomes
d'Hermite.
a
Hypergeometri-
Gauthier-Viliars, Paris.
Bora, S.L. and Kalla, S.L.C1970). Some results involving generalized function of two variables. Kyungpook
Math.
Bozhinov, N.(1988). Convolutional multipliers.
J.
10, 133-140.
representations
Sofia: Publ. House Bulg.
Acad.
of
commutants
and
Science.
Braaksma, B.L.J. (1963).Asymptotic expansions and analytic continua tions for a class of Barnes-integrals.Compositio Math.
Brychkov,Yu.A., Glaeske,H.-J. and Marichev,0.I. tion of Techniki. Soviet
integral Math.
Math.
VINITI.
(1983). Factoriza Nauki
i
21, 3-41.(Russian); English transl.in
J.
transformations of convolution
Anal.
15, 339-341.
type. Itogi
1985, 3.
Brychkov,Yu.A., (1992). Multidimensional
Glaeske,H.-J., integral
Prudnikov, transforms.
York, London, Paris, Montreux, Tokyo.
261
A.P.
and
Vu
Kim
Tuan.
Gordon and Breach, New
Burchnall, double
J.L.
and
hypergeometric
Chaundy,
T.W.(1940). Expansions Quart.
functions.
J.
Math.
of
Oxford
Appell's Ser.
11,
249-270.
Burchnall,
J.L.
and
Chaundy,
T.W.(1941). Expansions
(II). Quart.
double hypergeometric functions
J.
Math.
of
Appell's
Oxford
Ser.12,
112-128.
Jnanabha.
Buschman, R.G.(1977). H-functions of two variables, II. 20, 107-118.
Buschman, R.G.(1978). H-functions of two variables,I. Indian
J.Math.
20, 105-116.
Buschman, R.G. (1979a).H-functions of two variables, III.Pure Math.
Sci.
9, 13-18.
Buschman, R.G. (1979b). H-functions Math.
J.
of
N variables. Ranchi
Univ.
10, 81-88.
Buschman, R.G. Acad.
Appl.
Math.
(1981). H-function transformation chains. J.
Indian
3, N 2, 1-5.
Buschman, R.G.(1981a). Hindden reducibilities of H-functions of two variables. Indian
J.
Pure
Appl.
Math.
Buschman, R.G.(1981b).Reduction special
values of
the
variables.
12, 1448-1451.
formulas for Appell functions for Indian
J.
Pure
Appl.
Math.
12,
1452-1453.
Buschman,
R.G.
H-functions. Pure
Buschman, Acad.
Math.
(1982a).
Appl.
R.G.
Math.
(1982b).
Analytic Sci.
domains
for
multivariable
16, 23-27.
Factorization
4, 69-73.
262
of
H-functions.
J.
Indian
Buschman,
R.G.
relations. Ganita
(1987).
Buschman, R.G. Indian
J.
Math.
theorem
for
simple
contiguous
function
(1987a). Contiguous relations for Appell functions.
29, N 2, 165-171.
Buschman,
R.G.
G-function. Indian
Buschman,
A
1, N 1-2, 25-28.
(1987b).
J.
R.G.
Pure
Contiguous
Appl.
(1990).
Math.
Simple
relations
for
Meijer's
18, N 6, 536-547.
contiguous
function
functions defined by Mellin-Barnes integrals. Indian
J.
relations Math.
for
32, N 1,
25-32.
Buschman, R.G. and Gupta, K.C.(1975). Contiguous relations for the H-function
of
two
Indian
variables.
J.
Pure
Appl.
Math.
6, N 12,
1416-1421.
Buschman,
R.G.,
Koul,
C.L.
and
Gupta,
K.C.(1977).
Convolution Glasnik
integral equations involving the H-function of two variables. Matematicki
12, N 32, 61-66.
Buschman, R.G. and Srivastava, H.M.
(1975). Inversion formulas for
the integral transformation with the H-function as kernel. Indian Pure
Appl.
Math.
J.
6, N 6, 583-590.
Buschman, R.G. and Srivastava, H.M. (1986). Convergence regions for some multiple Mellin-Barnes contour integrals representing generalized hypergeometric functions. Internat.
J.
Math.
Ed. Sci.
Tech.
17, 605-609.
Chaturvedi, K.K. and Goyal, A.N.(1972). A -function. Indian Appl.
Math. Cih,
applications.
J.
Pure
3, 357-360. A.K.(1988).
The
multidimensional
residues
and
their
Nauka. Novosibirsk. (Russian).
Davis, B.(1978). Integral
transforms
New York: Springer-Verlag. 263
and their
applications.
Berlin.
Dimovski, I.H. (1974). A transform approach to operational calculus for
the general
Acad.
Bulg.
Sci.
Bessel-type
differential
operator.
Computers
rendus
27, N 2, 155-158.
Dimovski, I.H.(1981). Convolution representation of the commutant of Gel'fond-Leont'ev integration operator. C. R. Acad.
Bulg.
Sci.
34, N 12,
1643-1646.
Dimovski,
Convolutional
I.H.(1982).
calculus.
Sofia:
Publ.House
Bulg. Acad. Sci., 2. (Second ed. :Dorbrecht. Boston and London: Kluwer Acad. Publ.,East Europ. Ser. 1990, 43.).
Dimovski, I.H. and Kiryakova, V.S.(1983). Convolution and commutant of Gel'fond-Leont'iev operator of integration. In Proc. Constr.
Function
Theory'81
Intern.
Conf.
(Varna, 1981), Sofia: Publ. House Bulg. Acad.
Sci., 288-294.
Dimovski, differential Conf.
I.H.
and
property
Complex
Kiryakova,
V.S.(1984).
of Borel-Dzrbasjan
Analysis
and Appl.'81
Convolutions In Proc.
transform.
and Intern.
(Varna, 1981), Sofia: Publ. House
Bulg. Acad. Sci., 148-156.
Dimovski,
I.H.
and
Kiryakova,
V.S.(1985).
Transmutations,
convolutions and fractional powers of Bessel-type operators via Meijer's G-function. In Proc.
Intern.
Conf.
Complex
Analysis
and Appl.'S3
(Varna,
1983), Sofia: Publ. House Bulg. Acad. Sci., 45-66.
Dimovski,
I.H.
and
Hermite transform. Math.
Kalla, Japonica.
S.L.(1988).Explicit
Ditkin, V.A. and Prudnikov, A.P.(1958). Operational variables
and its
applications.
calculus.
for
calculus
of
two
Fizmatgiz. Moscow.(Russian).
Ditkin, V.A. and Prudnikov, A.P.(1965). Integral operational
convolution
33, N 3, 345-351.
Pergamon Press. 264
transformations
and
Ditkin, V.A. and Prudnikov, A.P. (1967). Integral transforms. Nauki
i Techniki.
Progress
Math.
in Math.
Anal.
VINITI.
Itogi
7-82.(Russian); English transl.in
1969, 4.
Dixon, A.L. and integrals. Quart.
J.
Ferrar, Math.
W.L.(1936).
Oxford
Doetsch, G.(1956). Handbuch
Ser.
A
class
of
discontinuous
7, 81-96.
der
Laplace-Transformation.
Basel und
Stuttgart: Birhauser Verlag.
Integral
Dzrbasjan, M.M. (1966). the
functions
in
the
complex
transforms
domain.
and
representations
of
Moscow. Nauka. (Russian).
Embrechts, P. and Omey, E.(1988). An Abelian theorem for a general class of Mellin type integral transforms. J.
Math.
Anal.
Appl.
132, N 1,
138-145.
Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G. (1953). Higher
Transcendental
Funct ions,
Vols I and II. McGraw-Hill, New York.
Hypergeometric
Exton,H.(1976).Multiple
Functions
and
Applications.
Ellis Horwood Ltd., Chichester, New York, Toronto.
Exton, H. (1978). Handbook
of Hyper geometric
Integrals.
Ellis Horwood
Ltd., Chichester, New York, Toronto.
Fedorjuk, M.V. (1986). VINITI
Integral
transforms. Itogi
Nauki
i
Techniki.
13, 211-253.(Russian).
Flensted-Jensen, M. and Koornwinder, T. structure for Jacobi function expansions. Ark.
(1973). The convolution Mat.
11, 245-262.
Fox,C.(1961). The G- and H-functions as symmetrical Fourier kernels. Tran.Amer.
Math.
Soc.
98, 395-429.
265
Glaeske, Math.
H.-J.(1981). Die
Aequat.
Laguerre-Pinney-Transformation.
22, 73-85.
Glaeske, H.-J.(1982). On a class of integral transformations. Ztsch.
F.-Schiller-Univ.
Glaeske, Tec.
Ing.
Jena.
Math.-Nat.
H.-J.(1986).On
Univ.
Zulia.
the
Wiss.
31, 579-586.
Wiener-Laguerre
transformation.
Rev.
9, 27-35.
Glaeske, H.-J.(1987).Operational properties of a generalized Hermite transformation. Aequat.
Math.
32, 155-170.
Glaeske, H.-J. and HeB, A.(1986). A convolution connected with the Kontorovich-Lebedev transform. Math.
Z.
193, 67-78.
Glaeske, H.-J. and HeB, A. (1987).On the convolution theorem of the Mehler-Fock-Transform Nachr.
for a class of generalized functions
(I). Math.
131, 107-117.
Glaeske, H.-J. and HeB, A. (1988).On the convolution theorem of the Mehler-Fock-Transform for a class of generalized functions (II). Math. Nachr.
136, 119-129.
Goldberg,
R.R.(1957).Convolution
functions. Riv.
Mat.
Goldberg, integrals. J.
Univ.
R.R.(1961). Math.
Anal.
transforms
of
almost
periodic
Parma 8, 307-312.
Spaces
Appl.
of
convolutions
and
fractional
3, 336-343.
Goyal, S.P.(1971). A generalized function of two variables I. Studies
Math.
N 1, 37-46.
Goyal, S.P.(1975). The H-function of two variables. Kyungpook J.
15, 117-131.
J.
Austr.
Gupta,
Univ.
K.C. Math.
and Soc.
Mittal,
P.K.(1970).
11, 142-148. 266
The
H-function
Math.
transform,
Gupta, K.C. and Mittal, P.K.(1971). The H-function transform,11. J. Austr.
Math.
Soc.
12, 444-450.
Hirchman, I.I. and Widder, D.W. (1955). The convolution
transform.
Princeton, New Jersy: Univ. Press.
Horn,
J.(1889).
Ueber
die
convergenz
Reihenzweier und dreuer veranderlichen. Math.
Ann.
der
hypergeometrichen
34, 544-600.
Horn, J.(1931). Hypergeometriche funktionen zweier veranderlichen.
Math. Ann. 105, 381-407. Joshi, C M . generalized
and Prajapat, M.L. (1977). On some results concerning
H-function
of
two variables. Indian
J.
Pure
Appl.Math.
8,
N 1, 103-116.
Kakichev, V.A.(1967). On the convolutions for integral transforms. Izv.
Acad.
Navuk.
BSSR.
Ser.
Fiz.-Mat.
Navuk.
N 2, 48-57.(Russian).
Kakichev, V.A.(1990).On the matrix convolution for power series.Izv. Vuzov.
Matematika.
2, 53-62.(Russian).
Kalla, S.L.(1969a). Integral operators involving Fox's H-function. Act a Mexicana
Cienc.
y Tecnol.
3, 117-122.
Kalla, S.L.(1969b). Integral operators involving Fox's H-function. II. Notas
Cie.
7, 72-79.
Kalla,S.L.(1972). On the solution of an integral equation involving a kernel of Mellin-Barnes type integral. Kyungpook
Kapoor, Cambridge
V.K.(1968).
Philos.
Soc.
On
Math,
a generalized and Phys.
Sci.
Math.
Stieltjes
J.
12, 93-101.
transform.
Proc.
64, 407-412.
Kapoor, V.K. and Masood, S.(1968). On a generalized L-H transform. Proc.
Cambridge
Philos.
Soc.
Math,
and Phys. 267
Sci.
64, 399-406.
Kesarwani,
R.N.(1962).The
kernels. I. Proc.
Amer.
Math.
G-functions
Soc.
as
Kesarwani, R.N.(1963a). The G-functions kernels.II. Proc.
Kesarwani,
Amer.
Math.
Soc.
R.N.(1963b).The
kernels. III. Proc.
Kesarwani,
Amer.
R.N.(1965a).
transform. Portugal.
Math.
Math.
Kesarwani,
Soc.
as
unsymmetrical
The
G-functions
as
kernels
in
chain
Fourier kernels.
115, N 3, 356-359.
Appl.
On
Math.
an
integral
Intern.
transform
involving
G-
20, 93-98.
V.S.(1989).Convolutions
integrals. In Proc.
Fourier
14, N 1, 271-277.
24, N 1, 39-45.
R.N.(1971).
function. SI AM J.
Kiryakova,
Soc.
Fourier
14, N 1, 18-28.
Kesarwani, R.N.(1965b). A pair of unsymmetrical Trans. Amer.
Fourier
as unsymmetrical
G-functions
Math.
unsymmetrical
13, N 6, 950-959.
Conf.
of
Complex
Erdelyi-Kober
Analysis
fractional
and Appl.'Sl
(Varna,
1987), Sofia: Publ. House Bulg. Acad. Sci., 273-283.
Kratzel,
E.C1965).
Eine
Meijer-Transformations. Wiss. Reihe.
Veralllgemeinerung Zeitschrift
der
der
FSU Jena.
LaplaceMath.
und
-Naturwiss.
14, N 5, 369-381.
Luchko, Yu.F and Yakubovich, S.B.
(1991). Generating operators and
convolutions for some integral transforms. Dokl.
Akad.
Nauk.
BSSR.
35,
N 9, 773-776. (Russian).
Luke, Y.L.C1969). The Special
Functions
and
Their
Approximations.
Vols.I and II. Academic Press . New York.
Marichev,
0.1.
(1974).
One
class
of
integral
convolution type with special functions in kernels. Izv. BSSR.
Ser.
Fiz.-Mat.
Navuk,
N 1, 126-127 (Russian). 268
equations Acad.
of
Navuk.
Marichev, O.I. (1976). Some integral equations of convolution type with
special
Fiz.-Mat.
functions
Navuk,
in
Izv.
kernels.
Acad.
Navuk.
BSSR.
Ser.
N 6, 119-120 (Russian).
Marichev, 0.1. (1978). Integral operators with special functions in the kernels which generalize Izv.
Akad.
Navuk
BSSR,
Ser.
integration operators of complex order.
Fiz.-Mat.
Navuk,
Marichev, 0.1. (1983). Handbook transcendental
functions,
theory
of
and
N 2, 38-44 (Russian).
integral
transforms
algorithmic
tables.
of
higher
Ellis Horwood
Ltd., Chichester, New York, Toronto. Marichev,
0.1.
(1983a).
hypergeometrie type. Vestsi
Asymptotic
Akad.
Navuk
behavior
BSSR,
Ser.
of
functions
Fiz.-Mat.
of
Navuk,
N 4,
and
their
18-25 (Russian). Marichev, 0.1. (1990). The hypergeometric some
applications
to
integral
to the degree of Doctor
type
and differential
science
(Dr.sc).
functions
equations.
Dissertation
Minsk, USSR,
Byelorussian
State Univ. 373p. (Russian).
Marichev, 0.1. and Vu Kim Tuan.(1983a).The definition of the general G-function
of
two
variables,
equations. Differentzialnye
its
particular
Uravnenija.
cases
and
differential
19, N 10, 1797-1799. Dep.
in
VINITI 11.11.82, N 687-83. 23 pp.(Russian).
Marichev, 0.1. and Vu Kim Tuan.(1983b).The problems of definitions and symbols of G- and H-functions of several variables. Rev. Ingr.
Univ.
Zulia
Teen.
Fac.
6, 144-151.
Marichev, 0.1. and Vu Kim Tuan.(1985). Composition structure of some integral transforms of convolution type. Proc. Vekua.
Inst.
Appl.
Math,
of
I.N.
Tbilisi. 1, N 1, 139-142 (Russian).
Marichev,
0.1.
and
Vu
Kim
Tuan.(1986).
Factorization
G-transformation
in two classes of functions. In Proc.
Complex
and
Analysis
Appl.'85
Intern.
of Conf.
(Varna, 1985), Sofia: Publ. House Bulg.
Acad. Sci., 720-735. 269
Mathai, A.M.
Functions
and Saxena, R.K.
with
Applications
in
Generalized
(1973).
Statistics
and
Hypergeometric
Physical
Sciences.
Springer-Verlag Lecture Notes No.348, Heildelilerg.
Mathai,
A.M.
applications
in
and
Saxena,
statistics
R.K.
The
(1978).
and other
H-functions
disciplines.
with
New York; London;
Sydney: John Wiley.
Meijer, G m n (z).
C.S.(1941).
Nederl.
Akad.
Multiplikations
Wetensch.
Proc.
die
Funktion
44, 1062-1070 = Indag.
theoreme
fur
Math.3,
pq
486-490. Meijer, C.S.(1946). On the G-function. I, II, III, IV, V, VI, VII, VIII. Proc.
Nederl.
Akad.
Wet.
49, 227-237, 344-356, 457-469, 632-641,
765-772, 936-943, 1063-1072, 1165-1175.
Mittal,
P.K.
and
Gupta,
K.C.U972).
generalized function of two variables. Proc.
An
integral
Indian
Acad.
involving
Sci.
Sect.
A
75, 117-123.
Mourya,
D.P.(1970).
Analityc
continuations
hypergeometric functions of two variables. Indian
J.
of Pure
generalized Appl.
Math.
1,
464-469.
Munot, P.C. and Kalla, S.L.(1971) On an extension of generalized function of two variables. Univ.
Nac.
Nguyen Thanh Hai.(1990a). On
the convergence
double hypergeometric series. Vestsi Navuk.
N 2,122; Dep. in VINITI
Tucuman Rev.
Acad.
Navuk.
Ser.
A 21, 67-84.
of Kampe de Feriet BSSR.
Ser.
Fiz.-Mat.
20.04.89, N 2582-V89.12 pp.(Russian).
Nguyen Thanh Hai.(1990b).On the theory of general Fox's H-function of two variables. Dokl.
Akad.
Nauk.
BSSR.
270
34, N 4, 297-300 (Russian).
Nguyen Thanh Hai.(1990c). The two variables degree
of
Candidate
and two-dimensional Candidate
of
integral
(Ph.D.).
Dissertation
functions
Minsk
hypergeometrictype
transforms.
Dissertation to the
pp.; The
USSR,141
of
Abstract
of
Minsk USSR. Byelorussian StateUniv. Press. 14 pp.
(Russian).
Nguyen Thanh Hai.(1992). On the criterion of clarification for the convergence of Mellin-Barnes type double integrals. Vestsi BSSR,
Ser.
Fiz.-Mat.
Navuk.
Akad.Navuk.
N 1, 25-31.(Russian).
Nguyen Thanh Hai and Buschman, R.G. (1992). Reduction the H-functions of two variables. J.
Indian
Acad.
Math,
formulas
for
(to appear).
Nguyen Thanh Hai, Marichev, 0.1. and Buschman, R.G.(1992) Theory of general H-functions of two variables. Rocky
Mountain
J.Math.
Cto appear)
Nguyen Thanh Hai, Marichev, 0.1. and Srivastava, H.M.(1992). A note on
the
convergence
of
series. J. Math. Anal. Nguyen
Thanh
certain
families
of
multiple
hypergeometric
Appl. (to appear).
Hai
and
Yakubovich,
S.B.
(1990).On
certain
two-
Akad.
Nauk.
dimensional integral transforms of convolution type. Dokl. BSSR.
34, N 5, 396-398 (Russian).
Nieva Pino, M.E. and Kalla, S.L.(1977). Operadores de integracion fractional Mexicana
que
involucran
de Ciencia
y Tecnol.
la
function
H
de
dos
variables
I.
Acta
11, 31-34.
Nieva Pino, M.E. and Kalla, S.L.(1978). Operadores de integracion fractional Mexicana
que
involucran
de Ciencia
y Tecnol.
la function
H de
dos
variables
Nikiforov, A.F. and Uvarov, V.B.(1978). The principles of special Osier, Amer.
Math.
functions.
Acta
of
the
theory
Leibniz
rule.
Moscow: Nauka.
T.J.(1971). Month.
II.
12, 32-41.
Fractional
78, N 6, 645-649.
271
derivatives
and
Osier, T.J.(1972a). An integral analogue of Taylor's series and its use in computing Fourier transforms. Math.
Osier, T.J.(1972b).The Comput.26(120),
Comput.
Calcutta
Math.
integral analog of the Leibniz rule.
903-915.
Pathak, R.S.(1970). Some results involving Bull.
26, N 118, 449-460.
Math.
Soc.
G- and
H-functions.
62, 97-106.
Prasad, Y.N. and Singh, A.K. (1981). Infinite integration of certain Bull.
products associated with Buschman's H-function of two variables. Math.
Soc.
Sci
Math.
R.S.
Roumanie,
25, N 73, 65-73.
Prudnikov, A.P., Brychkov, Yu.A. and Marichev 0.1.(1986). and Series.
Vol.1:
Elementary
Functions;
Vol.ZiSpecial
Integrals
Functions.
Gordon
Prudnikov, A.P., Brychkov, Yu.A. and Marichev 0.1.(1989).
Integrals
and Breach, New York, London, Paris, Montreux, Tokyo.
and Series.
Vol.
3: More special
funct
ions.
Gordon and Breach, New York,
London, Paris, Montreux, Tokyo.
Prudnikov,
A.P.,
Brychkov,
Yu.A.
and
Marichev
evaluation of integrals and Mellin transform. Itogi Math.
Anal.
VINITI.
Rathie,
i
The
Techniki.
27, 3-146.(Russian).
N.(1989).
variables. Ganita
0.1. (1989a). Nauki
Reduction
Sandesh.
formulas
for
the
H-function
of
two
1, 42-46.
Reed, I.S.(1944). The Mellin type of double integral. Duke Math.
J.
11, 565-572.
Saigo, M. , Marichev 0.1. and Nguyen Thanh Hai. representations of Gaussian series
(1989).Asymptotic
F , Clausenian series
F and Appell
series F
and F near boundaries of their convergence regions.
Univ.
Rep.
Sci.
19, N 2, 83-90.
272
Fukuoka
Saigo, M. and Yakubovich, S.B. (1991). On the theory of convolution integrals for G-transforms. Fukuoka Univ. Sci. Samko, S.G., integrals
Kilbas, A.A.
and derivatives
and
and some of
Rep. 21, N 2, 181-193.
Marichev their
0.1.(1987).
applications.
Fractional
Minsk. Nauka i
Tekhnika. Saxena, R.K.(1966). An inversion formula for a kernel involving a Mellin-Barnes type integral. Proc. Amer. Math. Soc.17, N 4, 771-779.
Saxena,
R.K.(1971).
An
integral
associated
with
generalized
H-function and Whittaker functions. Acta Mexicanc Ci. Teen. 5, 149-154. Saxena, V.P.(1970). Inversion formulae to certain integral equations involving H-function. Portugal
Math. 29(1-2), 31-42.
Shah, M.(1973a). Application of Hermite polynomials for certain properties of Fox's H-function of two variables. Univ.Nac. Ser,
A
Tucuman Rev.
23, 165-178.
Shah, M.(1973b). On some applications related to Fox's H-function of two variables. Publ.
Inst.
Math.
(Beograd)
(N.S.)
16, N 30, 123-133.
Sharma, B.L.(1965). On a generalized function of two variables. I. Ann. Soc.
Sci.Bruxeles
Ser.
I 79, 26-40.
Singh, R.(1970). An inversion formula for Fox H-transform, Nath.
Acad.
Sci.
India.
Proc.
A 40, 57-64.
Slater, L.J.(1966). Generalized
Hypergeometric
Functions.
Cambridge
Univ. Press. London-New York. Sneddon, I.N.(1951). Fourier
transform.
Sneddon, I.N.(1972). The use of integral Hill. 273
New York. McGrayHill. transform.
New York. McGray
Bull.
Srinivasan, V.(1963). On the generalized Meijer transform. Acad.
Polon.
Sci.
Ser.
A. 11, N 7, 431-440.
Srivastava, H.M.(1972). A class of integral equations involving the H-function as kernel. Nederl.
Akad.
Wetensch.
Proc.
Ser.A
75 =
Indag.
Math. 34(3), 212-220. Srivastava, H.M. and Buschman, R.G.(1973). Composition of fractional integral operators involving Fox's H-function. Acta Mexic.
Cien.
Tecnol.
7, N 1-3, 21-28. Srivastava, H.M. and Buschman, R.G.(1976). Mellin convolutions and H-function transformations. Rocky Mountain J. Math. 6, 331-343.
Srivastava, H.M. and equations
with
special
Buschman, R.G.(1977).Convolution
function
kernels.
integral
New Delhi, Bangalore: Wiley
Eastern Ltd. Srivastava, H.M. and Daoust, M.C.(1972). A note on the convergence of Kampe de Feriet's double hypergeometric series. Math.
Nachr.
53,
151-159. Srivastava, H.M., H-functions
of
One
Gupta,
and
K.C.
Two Variables
S.P. (1982). The
and
Goyal,
with
Applications.
South Asian
Publishers. India. Srivastava, H.M. , Koul, C.L. and Raina, R.K.(1985). A class of convolution integral equations. J. Math. Anal. Srivastava, Hypergeometric
H.M.
Series.
and
Karlsson,
P.W.
Appl. (1985).
108, N 1, 63-72. Multiple
Ellis Horwood Ltd.,Chichester, New York, Toronto.
Ta Li.(1960). A new class of integral transforms. Proc. Soc.
Gaussian
11, N 2, 290-298.
274
Amer. Math.
Internat.
Tandon, 0.P.(1983). Some predictions. Tech.
J.
Math.
Ed.
Sci.
14, 665-666.
Tanno, Y.(1956). An inversion formula for convolution transforms. Kodai
Math.
Semin.
Repts.
8, N 2, 79-84.
Tanno, Y.(1966). On a class of convolution transforms. Tohoku J.
Math.
18, N 2, 156-173.
Tanno, Y.(1967). On a class of convolution transforms Math.
J.
II.
Tohoku
of
Fourier
19, N 2, 168-186.
Titchmarsh, Integrals.
E.G.
(1937).
Introduction
to
Theory
Oxford Univ. Press, Oxford.
Verma,R.U. (1971). On the H-function of two variables. II. An. Univ.
"Al.
I.
Cuza"
Iasi
Sect.
I a Mat.
Sti.
(N.S) 17, 103-109.
Vilenkin, N.Ya.(1958). Matrix elements of the indecomposable unitary representations
for
motions
group
of
generalized Mehler-Fock transforms. Dokl.
Vu Kim Tuan.(1985). Some questions the
hypergeometric
type
Candidate (Ph.D.).
of
functions.
Minsk,
USSR,
the
Lobachevski's
AN SSSR.
the
theory
Dissertation Byelorussian
space
and
118, 219-222.
and applications
of
to
of
State
the
degree
Univ.
U8p.
(Russian).
Vu
Kim
Tuan.(1986a).
Generalized
integral
convolution type in some space of functions. Compl.
Anal,
and Appl.'SS
transformations
In Proc.
Intern.
of Conf.
(Varna, 1985), Sofia: Publ. House Bulg. Acad.
Sci., 418-433.
Vu Kim Tuan.(1986b). On the theory of generalized integral trans forms in a certain function space. Dokl. transl. in Soviet
Math.
Dokl.
AN SSSR.
33 (1986), 103-106.
275
286, 521-524; English
Vu Kim Tuan.(1986c). On the factorization of the convolution type integral transform in space L . Dokl.
Vu
Kim
structure.
Tuan.(1987).
Integral
AN Arm. SSR.
transforms
1, 7-10.(Russian).
and
their
composition
Dissertation to the degree of Doctor science (Dr.sc). Minsk,
USSR, Byelorussian State Univ. 322p. (Russian).
Vu Kim Tuan.(1988). Some integral transform of Fourier convolution type. Dokl.
AN SSSR.
300, 521-525; English transl. in Soviet
Math.
Dokl.
37 (1988), 669-673.
Vu Kim Tuan, Marichev, 0.1. and Yakubovich, S.B.(1986). Composition structure
of
integral
transformations.
English transl. in Soviet
Watanabe,
Math.
Y.(1931).
Dokl.
Notes
Dokl.
AN SSSR.
286, 786-790;
33 (1986), 166-169.
on
the
generalized
derivative
of
Riemann-Liouville and its application to Leibnitz's formula. I and II. Tohoku
Math.J.
Widder,
34, 8-27, 28-41.
The
D.W.(1946).
Laplace
transform.
Fourier
integral
London.
Oxford
Univ.
Press.
Wiener, applications.
The
N.(1937).
and
certain
of
its
New York. Dover Publications.
Yakubovich,
General
S.B.(1987a).
Kontorovich-Lebedev
type
and some of
index their
integral
applications.
transforms
of
Dissertation to
the degree of Candidate (Ph.D.). Minsk, USSR, Byelorussian State Univ. 130p.(Russian).
Yakubovich, S.B.(1987b). Lebedev Akad.
transform
Nauk BSSR.
Yakubovich,
and
On
the
convolution
its applications
to
for
integral
Kontorovich-
equations.
Dokl.
31, N 2, 101-103 (Russian). S.B.(1990).
convolutions. Dokl.
Akad.
On
Nauk.
the BSSR. 276
constructive
method
of
integral
34, N 7 , 588-591.(Russian).
Yakubovich,
S.B. (1991).
On
type for G-transforms. Vestsi
the Akad.
integral Navuk.
convolutions
BSSR,
Ser.
of
Laplace
Fiz.-Mat.
Navuk
N 6, 11-16.(Russian). Yakubovich, S.B.(1992a). On one class of the integral convolutions. Vestsi
Akad.
Navuk.
Yakubovich
S.B.
Ser.
Mechanics",
Fiz.-Mat.
(1992b).
Republic's
convolution. Nonlinear
BSSR,
Navuk
About
some
collection
, N 2 (Russian).
generalizations "Mathematical
of
Laplace
Physics
and
Kiev.(Russian) (to appear).
Yakubovich, S.B. and Luchko, Yu.F.(1991a). Generalizations of the Leibniz rule on the integral convolutions. Dokl.
Akad.
Nauk.
BSSR.35,
N 2, 111-115 (Russian).
Yakubovich,
S.B.
and
Luchko,
Yu.F.(1991b).
The
evaluation
of
integrals and series with respect to indices (parameters) of hypergeometric functions. Proc. ISSAC91.
Bonn,
Intern.
15-17 July,
Symp.
on Symbolic
and
Algebraic
Comp.
1991, 271-280.
Yakubovich, S.B. and Luchko, Yu.F. (1991c). The generalizations of integral
analog
Mathematicae
of
the
Leibniz
rule
on
the
convolutions.
Extracta
6, N 2, 64-66.
Yakubovich, S.B. and Nguyen Thanh Hai (1991). Integral convolutions for H-transforms. Izv.
Yakubovich,
Vuzov.
S . B. ,
Matematika
Nguyen
Thanh
8, 72-79.(Russian).
Hai
and
Buschman,
Convolutions for H-function transformations. Indian
J.
Pure
R.G. Appl.
(1992). Math.
(to appear).
Yanushauskas, A.I. (1980).
Double
Series.
Nauka.
Novosibirsk.
(Russian).
Zemanian, A.H.(1965). Generalized
integral
McGray Hill.
277
transforms.
New
York.
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Author index
Agarwal, R.P. 5,12,102,261 Aizenberg, L.A. 57,261 Anguino M.E.F. de. 261 Appell, P. 3,116,261
Bailey, W.N. 5 Barnes, E.W. 1 Bora, S.L. 6,12,74,261 Bozhinov, N. 261 Braaksma, B.L.J. 3,261 Brychkov, Yu.A. 2,145,261,272 Burchnall, J.L. 4,5,262 Buschman, R.G. 8,9,10,11,12,13,22,45,132,262,263,271,274,277
Chaturvedi, K.K. 6,12,74,263 Chaundy, T.W. 4,5,262 Cih, A.K. 57,263
Daoust, M.C. 103,274 Davis, B. 263 Dimovski, I.H. 162,190,264 Ditkin, V.A. 163,264,265
279
Dixon, A.L. 12,132,265 Doetsch, G. 265 Dzrbasjan, M.M. 265
Embrechts, P. 265 Erdelyi, A. 1,4,24,41,43,65,84,93,112,113,116,163,217,218,259,265 Exton, H. 5,265
Fedorjuk, M.V. 265 Ferrar, W.L. 12,132,265 Flensted-Jensen, M. 162,265 Fox,C. 2,144,265
Gauss, C.F. 1 Glaeske, H.-J. 145,162,261,266 Goldberg, R.R. 266 Goyal, A.N. 6,12,74,263,266 Goyal, S.P. 3,8,11,266,274 Gupta, K.C. 3,7,8,11,12,71,132,263,266,267,270,274
HeB, A. 162,266 Hirchman, I.I. 162,267 Horn, J. 3,59,267
Joshi, C M . 267
280
Kakichev, V.A. 162,267 Kalla, S.L.
6,12,74,261,264,267,270,271
Kampe de Feriet, J. 4,103,113,261 Kapoor, V.K. 144,267 Karlsson, P.W. 5,86,274 Kesarwani, R.N. 144,268 Kilbas, A.A. 220,233,273 Kiryakova,
V.S. 190,264,268
Koornwinder, T. 162,265 Koul, C.L. 263,274 Kratzel, E. 268 Kummer, E.E. 1
Luchko, Yu.F. 233,268,277 Luke, Y.L. 1,2,92,268
Magnus, W. 265 Marichev, 0.I. 2,11,15,43,58,71,82,85,92,104,116,119,120,141,144, 217,219,222,230,233,261,268,269,271,272,273,276 Masood, S. 144,267 Mathai, A.M. 2,3,270 Meljer, C.S. 2,270 Mellin, Hj. 1 Mittal, P.K.
7,12,71,132,266,267,270
Mourya, D.P. 270 Munot, P.C. 6,12,74,270
281
Nguyen Thanh Hai
4.104.116.184,270,271,272,277
Nieva Pino, M.E. 271 Nikiforov, A.F. 271
Oberhettinger, F. 265 Omey, E. 265 Osier, T.J.
233,271,272
Pathak, R.S. 6,12,74,272 Pincherle, S. 1 Prajapat. M.L. 267 Prasad, Y.N. 272 Prudnikov, A.P.
Raina, R.K.
2.84,92,93.145,156,163,223,233,250,261,264,265,272
274
Rathie, N. 272 Reed, I.S. 55,272
Saigo, M. 116,272,273 Samko, S.G. 220,233,273 Saxena, R.K.
2,3,74,270,273
Saxena, V.P. 4,273 Shah, M. 6,12,74,273 Sharma, B.L. Singh,
5,12,102,273
A.K. 272
Singh, R. 132,273 Slater, L.J.
1,43,58,68,86,233,273
282
Sneddon, I.N. 162,225,273 Srinivasan, V. 274 Srivastava, H.M. 3,5,8,11,22,86,103,104,132,263,271,274
Ta Li 274 Tandon, O.P. 22,275 Tanno, Y. 275 Titchmarsh, E.G. 119,121,131,162,275 Tricomi, F.G. 265
Uvarov,
V.B.
271
Verma, R.U. 6,12,74,275 Vilenkin, N.Ya. 162,275 Vu Kim Tuan 11,13,71,120,132,145,222,261,269,275,276
Watanabe, Y. 233,276 Widder, D.W. 161,276 Wiener, N. 162,276
Yakubovich, S.B. 120,144,162,184,222,233,268,271,273,276,277 Yanushauskas, A.I. 109,116,277 Yuzakov, A.P. 57,261
Zemanian, A.H. 277
283
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Subject index
addition formula for gamma-function 233 additional probability's integral 93 Agarwal's G-function 5 algorithm for the evaluation of index integrals and series 257 Appell functions (series) 42,113,114,115
beta-function 41,168,169 beta-integral 193 Bessel function 93,98,99 binomial coefficients 238 Boltz's formula 86 Buschman's H-function of variables 8,9
characteristic of the H-function of two variables 69 Clausenian function 117 composition structure of the G-transform 153 composition structure of the H-transform 140,153 convolution for modified Meijer transform 213 convolution for modified Hankel transform 221 convolution for modified Stieltjes transform 215 convolution for Riemann-Liouville operators 209,210 cos-transform 149 criterion for convergence region of H-integral 13,30,39,40 contiguous relations 47,48
285
degenerate Tricomi function 93,100 Dixon's summation theorem 234 double hypergeometric type series 57 double Mellin-Barnes type integrals 9,170 double Mel1 in transform 54 double Mel1 in transform of the H-function 56 Dougall's formula 234
factorization property 163 factorizations of simplest operators 157 Fox's H-function 3,9,12 Fourier integrals 131 Fubini theorem 121
G-convolution 198,199 G-function of two variables 5,89 G-transform 142,144 G-transform with Gauss hypergeometric function 155,236 Gauss-Legendre product formula 230 Gauss's hypergeometric series
1
Gaussian summation theorem 68,233 general beta-function 193 general hypergeometric function 1 general integral convolution 170 general Laplace convolution 190-193 generalized binomial coefficients 239
286
H-function of two variables 6,7,8,9 H-function of N variables 11 H-function with the third characteristic 72 H-integral 12 H-transform 132 Hankel transform 148,221 Hartogs theorem 51 Horn's list functions 43-46
identical transform 137 incomplete double series 64 index of the G-transform 143 index of the H-transform 133 integral analogs of Leibniz rules 250-252 integral convolutions 161 integral exponential function 93,95 integral transforms with Kummer and Whittaker functions 152 inverse G-transform 143 inverse Mel1 in transform 120 inverse H-transform 136 inverse operator to the operator with Kummer function 236 Jakobi polynomials 259
Kampe de Feriet function (series) 4,103,203 Kummer function 152
287
Laplace convolution 163 Lebesgue theorem 242 Leibniz rules 233,236-241,250-252
Macdonald function 93,99 max-convolution 186,187 Meijer G-function 2 Meijer transform 151,152,213 Mellin-Barnes type of contour integral 2,3 Mellin correspondences 159-161 Mellin-Parseval formula 119 Mellin transform 119,120 Mellin transform of the H-function 134 Mellin type convolution transform 119 Mittal's and Gupta's H-function 7 modified H-convolution 225 modified Laplace transform 146 UJf^L)-space 120 3JT1 (L)-space 121
non-complete gamma-function 93,98 operator with Tricomi function and its inverse 235 operator with algebraic function in the kernel 235 operator with generalized Laguerre polynomial 235
Pochhammer symbol 1,103
288
reduction formulas 80,83-89 Riemann-Lebesgue lemma 121 Riemann-Liouville integro-differential operators 147
Saalschutz's summation theorem 234 series representations 57,217,218 Sharma's S-function 5 sin-transform 149 Slater's notation 43,106 Slater's theorem 58,63,141 Sonine integral 224 Stieltjes transform 151,215 Stirling's formula 110,244 sum and maximum properties 82,93 sum-convolution 183,185
Tuan's generalization of the H-transform 132
Whittaker function 152
289
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NOTATIONS
(a) 1,103 8(s,t) 193 C k 238 n
[ na I 239
I J
(c\/) 143 f*(s) 119,120,121 (f*g)(x) 162,170 L
(f*g)(x) 163 (f*g) (x) 183,185 +
(f*g)
(x) 186
max
(a,/3) (f*g)(x) 211 K V
(f*g)(x) 214 S a (f*g)(x) 216
(a,0) (f*g)(x) 221 sin
(f*g)(x) 225 25 ( f ^ ) ( A , . ) ( x ) 2225
F [x,y; (a,a,A) ; (/3,b,B) ] 59 m n F (ct,p.jS'irj-x,-y) 44,113 291
F (a,j3,0' ;y,y'
;-x,-y) 44,115
F (a,a';£,£';y;-x,-y) 44,114 F (a,/3;y,y' ;-x,-y)
44,114
4
F (a,b;c;z) 152 F (a,b;c;z) 1 2 1
a , . . . ,a i P;
F p q L
b , . . . ,b i
q;
A:B;B' f
>
(a):(b);(b');
4,103
x,y C:D;D' [ ( c ) : ( d ) ; ( d ' ) ;
(a ))
m, n
p.q
z
P q
2 y
:::-[H
p.q
(x) 142
m (3) r(a( 1 ^): r(a( 2 ^ ); ( (a ^ )A |
m ,n :m ,n ;m ,n 1 1 2 2 3 3 p , a : p , q : p ,q x,y *1 n l *2 2 3 3
91
(P ( 1 > ): ( P < 2 ) ) ; ((3<3)) q
q
1
G (a,0,0* ;x,y) m ,n p.q I v
q
2
65
(a ,a ) p
P
' Oq ,b )J qy
292
3
.((a,a)
tfm'n p>q
1,p
O.b) l,q
o["f(u)j(x) 132
y
H[x,y; (a,a,A) ; (0,b,B) ; L J m
n
,LJ
s
t
H[x;(a,a) ;(0,b) ;L ] 9 m n s f
m , n : in , n ; m , n l 1 2 2 3 3 x,y p ,q :p ,q ;p ,q *1
1
2 ^2 *3 ^3
(1), f
(1)
(a
,a
(2)
): (a
(2), .
,a
H (a;r;-x,-y)46H (a;yt5;-x,-y)46 l"(x~af(x)) 147
f (2v£)U|f (u)| 148
}•
f(u)
151
2K (2V^)|o|f(u)l 151
rl" * l ^ ^ a i b j - x j U r f (u)l 152
r\l/2-p-v,l/2-p+v~\e{2x)
xak(x)|o["f(u)l
,a
(/3(1),b(1)):0(2),b(2));(|3(3),b(3))
H (a,0;y;-x,-y) 44,65
r(p)(i+xrpU
(3) ( 3 ) ^
); (a
W
^)}°[ f (u)
145
293
152
) 72
- 1
a
jx k(x)| offCu)! 145
(xVx" a )(f(x)) 146 (xaA;ax"a)(f(x)) 146
( x a * V a f ) ( x ) 235 (xaAl x"af)(x) 235
v
(x a L A x" a f)(x) 235 k
, afr(b)
(X
r
b|
\fUT F a)
X
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236
(x°VC .x~af)(x) 236 a, b
m jf(x,y); s,t| = f*(s,t) 54
an" ( D 120
m'1
( D 121
L(
t
L(x"1/2;0,co) 185 L , Ln 52 C L L , L. 9,10 s t (Lf)(x) 163 (Sf)(x) 220
j—sin(2\6<)|o['f(u)j 149
294
i—cos(2Vx)lo|f (u)j
149
a ,...■, a i
P
1
q
43,106
#4(a,/3;r,S;-x,-y) * (a»|3;y;-x,-y)
44
45
*2(0,0' ;r;-x,-y)
45
* C p ; y ; - x , - y ) 45 ^(a.jSjy,*' ;-x,-y) *A0C>7>V';-x,-y)
45
46
E ( a , a * , £ ; y ; - x , - y ) 46 J a , / 3 ; r ; - x , - y ) 46 (K,JLI)
133,190
6(T)
8,9
*(x)
3
*(x) 2
295