
doi: 10.24033/asens.1749
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 \(t\to 1\).
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
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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