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A Simple Proof of an Aomoto-Type Extension of Askey's Last Conjectured Selberg q-Integral

A simple proof of an Aomoto-type extension of Askey's last conjectured Selberg \(q\)-integral
Authors: Kevin W. J. Kadell;

A Simple Proof of an Aomoto-Type Extension of Askey's Last Conjectured Selberg q-Integral

Abstract

In 1944 A. Selberg evaluated a multivariable beta integral known as Selberg's beta integral. In [SIAM J. Math. Anal. 11, 938-951 (1980; Zbl 0458.33002)] \textit{R. A. Askey} conjectured a number of \(q\)-extensions of this integral based upon various \(q\)-beta integrals. One of these conjectured \(q\)-integrals was proved by the author in [SIAM J. Math. Anal. 19, No. 4, 969-986 (1988; Zbl 0643.33004)] and by \textit{L. Habsieger} [SIAM J. Math. Anal. 19, No. 6, 1475-1489 (1988; Zbl 0664.33001)]. In this paper the author proves an Aomoto-type extension of an other conjectured Selberg \(q\)-beta integral. Part of the proof is based on an argument by \textit{M. E. H. Ismail} [Proc. Am. Math. Soc. 63, 185-186 (1977; Zbl 0351.33002)]. The result generalizes Askey's last conjectured Selberg \(q\)-beta integral which was proved by \textit{R. J. Evans} in [Contemp. Math. 166, 341-357 (1994; Zbl 0820.33001)]. Other extensions of Selberg's beta integral were obtained and proved by \textit{R. A. Gustafson} in [Bull. Am. Math. Soc. 22, No. 1, 97-105 (1990; Zbl 0693.33001); SIAM J. Math. Anal. 23, No. 2, 525-551 (1992; Zbl 0764.33008)].

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Keywords

Aomoto-type extensions, Selberg's integral, Applied Mathematics, Other basic hypergeometric functions and integrals in several variables, Selberg \(q\)-integrals, q-transportation theory for the root system An−1, Aomoto-type extension, Analysis

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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.
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2
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