PROOF OF A q-ANALOG OF A CONSTANT TERM IDENTITY CONJECTURED BY FORRESTER

Doron Zeilberger*

Appeared in: J. Comb. Theory Ser. A 66(1994), 311-312.

As always in q-theory, (X; Q)_{n} will stand for the product (1 − X)(1 − QX)...(1 − Q^{n−1}X), and when the ”base” Q is q, we will abbreviate (X; q)_{n} to (X)_{n}. For any Laurent polynomial f in x_{1}, ..., x_{n}, CT (f) denotes the coefficient of x_{1}^{0}..x_{n}^{0}. Throughout this paper t := q^{a , s=}q^{b} , u = q^{c}.

Theorem( q-extension of (4.2) in [F2]):

CT ∏_{i=1}^{n}(1 + yx_{i}) ∏_{i=1}^{n} (x_{i})_{b}(\frac{q}{x_{i}})_{c} ∏_{1<=i<j<=n} (\frac{x_{i}}{x_{j}})_{a}(\frac{qx_{j}}{x_{i}})_{a} =  (∗)

∏_{j=0}^{n−1} \frac{(q)_{b+c+ja}(q)_{(j+1)a}}{(q)_{b+ja}(q)_{c+ja}(q)_{a}} ∑_{r=0}^{n} \frac{(t; t)_{n}(u; t)_{r}(qs; t)_{n−r}}{(t; t)_{r}(t; t)_{n−r}(qs; t)_{n}} (−qy)^{r} ,

Proof: As in [Z], first use the Stembridge-Stanton trick, to transform (*) to the equivalent ”antisymmetric” version, let’s call it (*’), in which (\frac{qx_{j}}{x_{i}})_{a} is replaced by (\frac{qx_{j}}{x_{i}})_{a−1}, and the right side of (*) gets multiplied by (1 − t)/(t; t)_{a}. Next expand the very first product on the left of (*’), into a sum of 2^{n} terms, and note that they are all bad guys (see [Z], p. 314 ), except for the n + 1 terms x_{1}...x_{r}y^{r}, r = 0, ..., n, the corresponding constant terms of which are evaluated by [Z]’s eq. (5.1). QED

Historical Notes: The special case q = 1, b = 1,c = 0, of the above theorem was conjectured in [F1]. Shaun Cooper[C] formulated a conjecture for the general q-case, with still b = 1, c = 0. After Peter Forrester received a preliminary version of the present paper, that proved Cooper’s conjectured q-extension of his original conjecture, he also received the preprint [K1], from which[F2] he was able to derive the special case q = 1 of the above theorem. Forrester told us that the general theorem should follow from [K2] in an analogous way.

The present proof is shorter (even with [Z]) than Kaneko’s proof, and entirely elementary, but Kaneko proves an even more general theorem.

Acknowledgement: Many thanks are due to Shaun Cooper and Peter Forrester for helpful remarks.

References:

[C] S. Cooper, private communication.

[F1] P.J. Forrester, Some multidimensional integrals related to many body systems with the 1/r^{2} potential, preprint, La Trobe University, Bundoora, Victoria, Australia.

* Department of Mathematics, Temple University, Philadelphia, PA19122, USA; zeilberg@euclid.math.temple.edu. Supported in part by NSF grant DMS8901610.

[F2] ”, Selberg correlation integrals and the 1/r^{2} quantum many body system,preprint, La Trobe University, Bundoora, Victoria, Australia.

[K1] J. Kaneko, Selberg integrals and hypergeometric functions associated with Jack polynomials, preprint.

[K2] ”, q-Selberg integrals and hypergeometric functions associated with Macdonald symmetric polynomials, in preparation.

[Z] D. Zeilberger, A Stembridge-Stanton style elementary proof of the Habsieger-Kadell q-Morris identity, Discrete Math. 79(1989/90) 313-322. Jan 18, 1992. Revised: Jan 31, 1992 ; June 9, 1992.