1996
DOI: 10.24033/asens.1749
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$q$-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.

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Cited by 73 publications
(85 citation statements)
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References 14 publications
(26 reference statements)
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“…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.…”
Section: Introductionmentioning
confidence: 98%
“…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.…”
Section: Introductionmentioning
confidence: 98%
“…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).…”
Section: More Basic Hypergeometric Identities Involving Macdonald Polmentioning
confidence: 99%
“…(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.…”
Section: Introductionmentioning
confidence: 99%
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