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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
P1: GRNP1: GRN THE RAMANUJAN JOURNAL
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A 191 SUMMATION THEOREM FOR MACDONALD POLYNOMIALS
381
[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 −∞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
