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THE RAMANUJAN JOURNAL 2, 379Ð386 (1998) c 1998 Kluwer Academic Publishers. Manufactured in The Netherlands. °
A 1 Ψ1 Summation Theorem for Macdonald Polynomials JYOICHI KANEKO
[email protected] Graduate School of Mathematics, Kyushu University, Ropponmatsu, Fukuoka 810-8560, Japan Received November 13, 1996; Accepted March 16, 1998
Abstract. In this note we extend the RamanujanÕs 1 ψ1 summation formula to the case of a Laurent series extension of multiple q-hypergeometric series of Macdonald polynomial argument [7]. The proof relies on the elegant argument of Ismail [5] and the q-binomial theorem for Macdonald polinomials. This result implies a q-integration formula of Selberg type [3, Conjecture 3] which was proved by Aomoto [2], see also [7, Appendix 2] for another proof. We also obtain, as a limiting case, the triple product identity for Macdonald polynomials [8]. Key words: Macdonald polynomials, q-hypergeometric functions, 191 summation, q-integration formula of Selberg 1991 Mathematics Subject ClassiÞcation:
1.
PrimaryÑ33D20
Introduction
Fix q with 0 < |q| < 1 and set, for complex a, n, (a)∞ = (a; q)∞ =
∞ Y (1 − aq i ), i=0
(a)n = (a)∞ /(aq )∞ . n
The RamanujanÕs 1 ψ1 summation formula is (cf. [1, 4, 5]) ∞ X (ax; q)∞ (q/ax; q)∞ (q; q)∞ (b/a; q)∞ (a; q)n n x = , (x; q)∞ (b/ax; q)∞ (b; q)∞ (q/a; q)∞ n=−∞ (b; q)n
(1.1)
where |b/a| < |x| < 1. In this note we extend this to the case of a Laurent series extension of multiple q-hypergeometric series of Macdonald polynomial argument [7]. We Þrst recall the deÞnition of the q-hypergeometric series. Let Pλ (z 1 , . . . , z n ; q, t) be a Macdonald polynomial corresponding to a partition λ = (λ1 , . . . , λn ) of length (:= the number of parts) ≤ n [9, VI]. Note that when t = q, Pλ (z; q, q) is the Schur polynomial sλ (z). Consider the diagram of the partition λ in which the rows and columns are arranged as in a matrix, with the ith row consisting of λi squares. We denote the number of squares in the jth column by λ0j . for each square s = (i, j) in the diagram of λ, let a(s) = λi − j, l(s) = λ0j − i,
a 0 (s) = j − 1, l 0 (s) = i − 1,
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and put h λ (q, t) =
Y¡
¢ 1 − q a(s) t l(s)+1 ,
h 0λ (q, t) =
s∈λ
Y¡
¢ 1 − q a(s)+1 t l(s) .
s∈λ (q,t)
We deÞne the generalized factorial (a)λ (q,t)
(a)λ
by
Y ¢ 0 0 t l (s) − q a (s) a = t n(λ) (at 1−i )λi ,
Y¡
=
s∈λ
(1.2)
i≥1
where n(λ) =
X
(i − 1)λi =
i≥1
X λ0 (λ0 − 1) i i . 2 i≥1 (q,t)
Let a1 , . . . , ar and b1 , . . . , bs be complex numbers such that (bl )λ 6= 0, 1 ≤ l ≤ s for any (q,t) partition of length ≤n. The q-hypergeometric function r 8s (a1 , . . . , ar ; b1 , . . . , bs ; x), x = (x1 , . . . , xn ), is deÞned by (cf. [7]) (q,t) (a1 , . . . , ar ; b1 , . . . , bs ; x) r 8s
=
X λ
0 ªs+1−r (q,t) © (−1)|λ| q n(λ ) k=1 (ak )λ Qs (q,t) 0 h λ (q, t) l=1 (bl )λ
Qr
Pλ (x; q, t). (1.3)
We deÞne the Laurent series extension of this as follows. First note that for a partition λ of length ≤n we have [7, Proposition 3.2] Ã h λ (q, t) =
(t)n∞
n Y ¡ λi n−i+1 ¢ q t ∞
!−1
i=1
à h 0λ (q, t)
=
(q)n∞
n Y ¡ λi +1 n−i ¢ t q ∞ i=1
Y 1≤i< j≤n
!−1
Y 1≤i< j≤n
¡ λ −λ j−i+1 ¢ q i jt ¡ λ −λ ¢ ∞. q i j t j−i ∞
(1.4)
¡ λ −λ +1 j−i ¢ q i j t ∞ ¡ λ −λ +1 ¢ . q i j t j−i−1 ∞
(1.5)
Denote Zn≥ = {(λ1 , . . . , λn ) ∈ Zn | λ1 ≥ · · · ≥ λn }. For any λ ∈ Zn≥ we let h λ (q, t) and h 0λ (q, t) be deÞned by these formula. The generalized (q,t) factorial (a)λ is deÞned by the right most side of (1.2). Finally we notice that for a partition λ it holds that [9, VI, (4.17)] Pλ (x; q, t) = (x1 · · · xn )λn Pλ−λn (x; q, t),
(1.6)
where λ − λn := (λ1 − λn , . . . , λn−1 − λn , 0). So we let Pλ (x; q, t) be deÞned by this formula for any λ ∈ Zn≥ . Essential use will be made of the following evaluation formula
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[9, VI, (6.110 )]: (q,t)
Pλ (1, t, . . . , t n−1 ; q, t) =
(t n )λ . h λ (q, t)
(1.7)
This holds for any partition λ and hence, as can be easily checked, for any n-tuple in Zn≥ . Note that this evaluation is equivalent to the following q-binomial theorem for Macdonald polynomials [8, Remark, p. 359]: (q,t)
1 80
(a; −; x1 , . . . , xn ) =
n Y (axi ; q)∞ i=1
(xi ; q)∞
.
(1.8)
We now deÞne (q,t) r 9s+1 (a1 , . . . , ar ; b, b1 , . . . , bs ; x)
X
=
λ1 ≥λ2 ≥···≥λn −∞<λi <∞
X
=
λ1 ≥λ2 ≥···≥λn −∞<λi <∞
×
Y 1≤i< j≤n
0 ªs+1−r (q,t) © (−1)|λ| q n(λ ) k=1 (ak )λ Qs (q,t) 0 h λ (q, t) l=1 (bl )λ
Qr
¡ λ n−i ¢ n bq i t (q)n∞ Y ¡ ¢∞ Pλ (x; q, t) (b)n∞ i=1 q λi +1 t n−i ∞
0 ªs+1−r (q,t) © n Y (−1)|λ| q n(λ ) k=1 (ak )λ Qs Q (q,t) n n−i ) λi i=1 l=1 (bl )λ i=1 (bt
Qr
(bt n−i )∞ (b)∞
¡ λ −λ +1 j−i−1 ¢ q i j t ¡ λ −λ +1 ¢ ∞ Pλ (x; q, t). q i j t j−i ∞
(1.9)
Since (q λn +1 )∞ = 0 when λn < 0, we see (q,t) r 9s+1 (a1 , . . . , ar ; q, b1 , . . . , bs ; x)
= r 8(q,t) (a1 , . . . , ar ; b1 , . . . , bs ; x). s
(1.10)
The main result of this paper is Theorem.
Let |b/a| < |xi | < 1 for 1 ≤ i ≤ n. Then we have ( ) n Y (axi )∞ (q/axi )∞ (bt i−1 /a)∞ (q)∞ (q,t) . (a; b; x) = 1 91 (xi )∞ (b/axi )∞ (qt i−1 /a)∞ (b)∞ i=1
(1.11)
The proof relies on the elegant argument of Ismail [5] and the q-binomial theorem for Macdonald polynomials (1.8). The q = t case was proved by Milne [10], the proof of (q,q) which is based on a determinantal formula of r 9s+1 . We remark that the sum formula above implies a q-integration formula of Selberg type [3, Conjecture 3] which was proved by Aomoto [2], see also [7, Appendix 2] for another proof. We also obtains, as a limiting case, the triple product identity for Macdonald polynomials which we derived directly from the q-binomial theorem for Macdonald polynomials in [8]. Finally, it should be noted that the IsmailÕs argument has been utilized by Milne to prove his U (n) multiple series generalization of 1 91 summation, see [11] and references therein.
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KANEKO
Proof of main result (q,t)
We Þrst deal with the convergence of r 9s+1 . Let R = |bb1 · · · bs /a1 · · · ar |. Proposition. We have (1) If r < s + 1, then the series (1.9) converges absolutely for all |xi | > R, 1 ≤ i ≤ n. (2) If r = s + 1, then the series (1.9) converges absolutely for R < |x1 · · · xn |/(nkxk)n−1 , |xi | < n −1 , 1 ≤ i ≤ n, where kxk = max{|x1 |, . . . , |xn |}. (3) If r > s + 1, then the subseries of (1.9) of terms with λn ≥ 0 converges absolutely in a neighborhood of the origin only when it terminates. Proof:
It follows from [7, Lemma 3.7] that |Pλ (x)| ≤ C(nkxk)|λ| .
provided λn ≥ 0. Hence, in general |Pλ (x)| ≤ C|x1 · · · xn |λn (nkxk)|λ−λn | . We have for some constant C 0 that ¯Q ¯ r (a )(q,t) ©(−1)|λ| q n(λ0 ) ªs+1−r X ¯ k=1 k λ ¯ Qs (q,t) Qn ¯ (bl ) (bt n−i )λ λ ≥λ ≥···≥λ 1
2
n
−∞<λi <∞
l=1
λ
i=1
i
Y 1≤i< j≤n
¯ ¡ λ −λ +1 j−i−1 ¢ ¯ q i j t ¯ ¡ λ −λ +1 ¢ ∞ Pλ (x; q, t)¯ ¯ q i j t j−i ∞
¯ à ! Qr ∞ ¯ n−1 1−i Y X )λi k=1 (ak t (s+1−r )λi (λi −1)/2 (r −s)(i−1)λi λi Qs q t (nkxk) ≤C¯ 1−i ) (bt n−i ) ¯ i=1 λ =−∞ λi λi l=1 (bl t i Qr µ ¶ ¯ ∞ 1−n X )λn |x1 · · · xn | λn ¯¯ k=1 (ak t (s+1−r )λn (λn −1)/2 (r −s)(n−1)λn Qs × q t ¯. 1−n ) (b) ¯ nkxkn−1 λn λn l=1 (bl t λ =−∞ 0¯
n
Hence, the conclusions of the cases r ≤ s + 1 are immediate from (A)λi = (−1)λi q λi (λi −1)/2 Aλi (A−1 q)−1 −λi . The proof for the case of r > s + 1 is identical to that of [7, Theorem 3.8] and we omit it. We proceed to the proof of Theorem. Note that each side of (1.11) is an analytic function of b in a neighborhood of the origin. Hence it sufÞces to verify that both sides of (1.11) coincide when b = q N , N ∈ Z>0 . Since (bq λn )∞ = (q λn +N )∞ = 0 provided λn ≤ −N , we have ¡ λ +N N −i ¢ n Y X q i t (q,t) N ∞ (a)λ 1 91 (a, q ; x) = N) (q ∞ ∞>λ1 ≥···≥λn ≥−N +1 i=1 ¡ λ −λ +1 j−i−1 ¢ Y q i j t ¡ λ −λ +1 ¢ ∞ Pλ (x; q, t). × (2.1) j−i i j t q 1≤i< j≤n ∞
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Observe that (at1−i )λi = (aq −N +1 t 1−i )λi +N −1 (at1−i )−N +1 n(λ) = n(λ + N − 1) −
n(n − 1) (N − 1) 2
Pλ (x; q, t) = Pλ+N −1 (x; q, t)(x1 · · · xn )1−N , where λ + N − 1 = (λ1 + N − 1, . . . , λn + N − 1). Substituting these into (2.1), we get ( (q,t) (a, q N ; x) 1 91
=
t
−n(n−1)(N −1)/2
(q)nN −1
n Y (at 1−i )−N +1 (x1 · · · xn )1−N i=1
X
×
∞>λ1 ≥···≥λn ≥0
×
1≤i< j≤n
( =
Y
t
(q,t) (aq −N +1 )λ
¡ λ +1 n−i ¢ n Y q i t ∞ i=1
(q)∞
¡ λ −λ +1 j−i−1 ¢ q i j t ¡ λ −λ +1 ¢ ∞ Pλ (x; q, t) q i j t j−i ∞
−n(n−1)(N −1)/2
(q)nN −1
)
n Y (at 1−i )−N +1 (x1 · · · xn )1−N
)
i=1 (q,t) × 1 80 (aq −N +1 ; −; x)
(
=
n Y t −n(n−1)(N −1)/2 (q)nN −1 (at 1−i )−N +1 (x1 · · · xn )1−N
)
i=1
×
n Y i=1
(aq −N +1 xi )∞ . (xi )∞
(2.2)
On the other hand, we see (axi )∞ (axi )∞ (q/axi )∞ = (1 − q/axi ) · · · (1 − q N −1 /axi ) N (xi )∞ (q /axi )∞ (xi )∞ = (−1) N −1 q N (N −1)/2 (axi )1−N
(aq −N +1 xi )∞ , (xi )∞
(q N t i−1 /a)∞ = (−1) N −1 q N (1−N )/2 t (N −1)(1−i) a (N −1) (at 1−i )−N +1 (qt i−1 /a)∞ (q)∞ = (q) N −1 . (q N )∞
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Hence, ¾ n ½ Y (axi )∞ (q/axi )∞ (q N t i−1 /a)∞ (q)∞ (xi )∞ (q N /axi )∞ (qt i−1 /a)∞ (q N )∞ ( ) n n Y Y (aq −N +1 xi )∞ −n(n−1)(N −1)/2 n 1−i 1−N = t (q) N −1 (at )−N +1 (x1 · · · xn ) . (xi )∞ i=1 i=1
i=1
2
This completes the proof of our theorem. 3.
Consequences
Let 0q (x) be the q-gamma function and ϑ(x) the Jacobi elliptic theta function: 0q (x) = (1 − q)(1−x)
(q)∞ , (q x )∞
ϑ(x) = (x)∞ (q/x)∞ (q)∞ .
We deÞne a multiple q-integral by Z ∞ Z ∞ ··· f (t1 , . . . , tn )dq t1 · · · dq tn := (1 − q)n 0
X
¡ ¢ f q s1 , . . . , q sn q s1 +···+sn .
−∞<si <∞ 1≤i≤n
0
Corollary 1. We have ¡ ¢ µ 1−k ¶ Z ∞ Z ∞Y n Y −cti q a+b+2(n−1)k ∞ tjq a−1 ··· ti 1n (t) ti2k−1 dq t1 · · · dq tn (−cti )∞ ti 0 0 2k−1 i=1 1≤i< j≤n = n!q ×
¡ ¢ n ϑ −cq a+(2n−i−1)k − q k )n Y ¡ ¢ (q k ; q k )n i=1 ϑ −cq (n−i)k
ka ( n2 )+2k 2 ( n3 ) (1
n Y 0q (a + (n − i)k)0q (b + (n − i)k)0q (1 + ik)
0q (a + b + (2n − i − 1)k)0q (1 + k)
i=1
,
(3.1)
Q where 1n (t) = 1≤i< j≤n (ti −t j ). This is equivalent to the original nonsymmetrical version [3, Conjecture 3]: Z
∞
Z
∞
···
0
0
=q
n Y
¡ tia−1
i=1
kx( n2 )+2k 2 ( n3 )
−cti q a+b+2(n−1)k (−cti )∞
¢ ∞
Y 1≤i< j≤n
µ ti2k
t j q 1−k ti
¶ dq t1 · · · dq tn 2k
¢ n ¡ n Y ϑ −cq a+(2n−i−1)k Y 0q (a + (n − i)k)0q (b + (n − i)k)0q (1 + ik) ¢ · ¡ . (n−i)k 0q (a + b + (2n − i − 1)k)0q (1 + k) ϑ −cq i=1 i=1
(3.2)
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Proof:
385
Set t = qk, xi = q a+(n−i)k , a = −cq
i = 1, . . . , n,
(n−1)k
b = −cq a+b+2(n−1)k , in (1.11) and use (1.7). Then it is not difÞcult to verify (3.1). The nonsymmetrical form (3.2) follows from the observation of Kadell [6, p. 976]: Z ···
Z Y m i=1
Y
ti
(ti − q k t j )dµ (t) =
1≤i< j≤n
(q k ; q k )m (q k ; q k )n−m 1 (1 − q k )n m!(n − m)! Z Z Y m Y t1 (ti − t j ) dµ (t) × ··· i=1
1≤i< j≤n
where dµ (t) =
n Y
¡ tia−1
−cti q a+b+2(n−1)k
¢
(−cti )∞
i=1
Y
∞
µ ti2k−1
1≤i< j≤n
t j q 1−k ti
¶ dq t1 · · · dq tn . 2k−1
2
Corollary 2. Let z 1 z 2 · · · z n 6= 0. Then we have n Y ¡ ¢ (−z i q)∞ −z i−1 ∞ (q)∞ = 0 i=1
(
X
q
λ +1 λ +1 ( 1 2 )+···+( n 2 )
λ1 ≥λ2 ≥···≥λn −∞<λi <∞
¡ λ −λ +1 j−i−1 ¢ ) q i j t ¡ λ −λ +1 ¢ ∞ j−i i j t q 1≤i< j≤n ∞ Y
× P(λ1 ,...,λn ) (z 1 , . . . , z n ; q, t). Proof:
(3.3)
Replace a, b and xi with −1/c, 0 and qcz i respectively in (1.11) and observe that µ
t 1−i − c
¶(q,t) λ
(qc)|λ| = q (
λ1 + 1 2
λ +1 )+···+( n 2 )
n Y ¡
−ct i−1 q 1−λi
i=1
Then letting c → 0 immediately gives the desired identity.
¢ λi
. 2
References 1. G.E. Andrews, Òq-Series,Ó CBMS Regional Conference Series in Math., American Mathematical Society, Providence, RI, 1986. 2. K. Aomoto, ÒConnection formulas of the q-analog de Rham cohomology,Ó in Functional Analysis on the Eve of the 21st Century (S. Gindikin, J. Lepowsky, and R.L. Wilson, eds.), Birkhauser, Boston, 1995, vol. 2, pp. 1Ð12.
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3. R. Askey, ÒSome basic hypergeometric extensions of integrals of Selberg and Andrews,Ó SIAM J. Math. Anal. 11 (1980), 938Ð951. 4. G. Gasper and M. Rahman, Basic Hypergeometric Functions, Cambridge University Press, London, 1990. 5. M.E.H. Ismail, ÒA simple proof of RamanujanÕs 1 91 sum,Ó Proc. Amer. Math. Soc. 63 (1977), 185Ð186. 6. K. Kadell, ÒA proof of AskeyÕs conjectured q-analogue of SelbergÕs integral and a conjecture of Morris,Ó SIAM J. Math. Anal. 19 (1988), 969Ð986. « 7. J. Kaneko, Òq-Selberg integrals and Macdonald polynomials,Ó Ann. Sci. Ecole Norm. Sup., 4e s«erie, 29 (1996), 583Ð637. 8. J. Kaneko, ÒA triple product identity for Macdonald polynomials,Ó J. Math. Anal. Appl. 200 (1996), 355Ð367. 9. I.G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Oxford University Press, Oxford, 1995. 10. S.C. Milne, ÒSummation theorems for basic hypergeometric series of Schur function argument,Ó in Progress in Approximation Theory (A.A. Gonchar and E.B. Saff, eds.), Springer-Verlag, New York, 1992, pp. 51Ð77. 11. S.C. Milne, ÒThe multidimensional 1 91 sum and Macdonald identities for A(1) l ,Ó in Proceedings of Symposia in Pure Mathematics: Part II, Theta Functions Bowdoin 1987 (L. Ehrenpreis and R.C. Gunning, eds.), Amer. Math. Soc., 1989, vol. 48, pp. 323Ð359.
