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Article . 1996
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Annales Scientifiques de l École Normale Supérieure
Article . 1996 . Peer-reviewed
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$q$-Selberg integrals and Macdonald polynomials

\(q\)-Selberg integrals and Macdonald polynomials
Authors: Kaneko, Jyoichi;

$q$-Selberg integrals and Macdonald polynomials

Abstract

Summary: We consider a Jackson integral with special integrand (\(q\)-Selberg integral) and give an explicit formula for a system of \(q\)-difference equations satisfied by it. We also define a kind of hypergeometric function having series expansions in terms of Macdonald polynomials and show that this function satisfies a \(q\)-difference equation formed by summing up equations of the \(q\)-difference system above after multiplying each by a suitable factor. We can thus conclude the \(q\)-Selberg integral to be the hypergeometric function in our sense. This implies, in particular, the \(q\)-integration formula of Macdonald polynomials due to \textit{K. W. J. Kadell} [The Selberg-Jack symmetric functions, Adv. Math. 130, No. 1, 33-102 (1997; Zbl 0885.33009)]. These results reproduce our previous ones [\textit{J. Kaneko}, SIAM J. Math. Anal. 24, No. 4, 1086-1110 (1993; Zbl 0783.33008)] if we put \(q= t^\alpha\) and let \(t\to 1\).

Keywords

Generalized basic hypergeometric series, Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.), Symmetric functions and generalizations, \(q\)-calculus and related topics, \(q\)-Selberg integral, Macdonald polynomials, Connections of hypergeometric functions with groups and algebras, and related topics

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
42
Top 10%
Top 10%
Top 10%
bronze