$q$-Selberg integrals and Macdonald polynomials

Abstract

We prove a generalization of the q-Selberg integral evaluation formula. The integrand is the q-Selberg integral's one multiplied by a factor of the same form with respect to part of the variables. The proof relies on the norm formula of Koornwinder polynomials. We also derive generalizations of Mehta's integral formula as limit cases of our integral.

q-Selberg Integrals and Macdonald Polynomials

“…By replacing q β → −q 1−α−2(n−1)k /c and then letting c → −q β in Theorem 1.1, and by noting that (x i q; q) ∞ = 0 for x i = q ki with k i a negative integer, we obtain as second corollary another generalized q-Selberg integral. This integral was first conjectured by Kadell [11,Conjecture 8], and proved for k = 1 by Kadell [11] and for general k by Kaneko [16,Proposition 5.2] and Macdonald [20,Example VI.9.3]. See also Kadell [14, Theorem 1] for the q → 1 limit.…”

Macdonald Polynomials and Multivariable Basic Hypergeometric Series

“…While nonterminating series are considered in Section 7, one can ask whether other important basic hypergeometric identities can be extended to the multivariate setting involving A n−1 Macdonald polynomials. Concerning multivariate extensions of identities for non-very-well-poised basic hypergeometric series, we refer to the work of Kaneko [23,24], Baker and Forrester [3], and Warnaar [54], where several identities are established that involve Macdonald polynomials playing the role of the argument of the respective series. In the very-well-poised case, which is investigated here, the A n−1 Macdonald polynomials play the role of q-shifted factorials (to which they would reduce after principal specialization).…”

“…(4)], Kaneko [23,24], Baker and Forrester [3], and Warnaar [54]. These authors in fact derived multivariable analogues of many of the classical summation and transformation formulae for basic hypergeometric series.…”

“…(4)] obtained a q-binomial theorem for multiple series of Macdonald polynomial argument. This identity became in fact the starting point in the development of a whole theory of identities for series of (classical A n−1 ) Macdonald polynomial argument, see Kaneko [23,24], Baker and Forrester [3], and Warnaar [54].…”

“…I would also like to express my sincere gratitude to the organizers of the "Workshop on Jack, Hall-Littlewood and Macdonald Polynomials" (ICMS, Edinburgh, September [23][24][25][26] 2003) for inviting me to participate in that very stimulating workshop. Among them, I am especially indebted to Vadim Kuznetsov whose interest in explicit formulae for Macdonald polynomials served as an inspiration for the present work.…”