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July 2014 Ramanujan’s $_{1}\psi_{1}$ summation theorem —perspective, announcement of bilateral $q$-Dixon–Anderson and $q$-Selberg integral extensions, and context—
Masahiko Ito, Peter J. Forrester
Proc. Japan Acad. Ser. A Math. Sci. 90(7): 92-97 (July 2014). DOI: 10.3792/pjaa.90.92

Abstract

The Ramanujan $_{1} \psi_{1}$ summation theorem is studied from the perspective of Jackson integrals, $q$-difference equations and connection formulae. This is an approach which has previously been shown to yield Bailey’s very-well-poised $_{6} \psi_{6}$ summation. Bilateral Jackson integral generalizations of the Dixon–Anderson and Selberg integrals relating to the type $A$ root system are identified as natural candidates for multidimensional generalizations of the Ramanujan $_{1} \psi_{1}$ summation theorem. New results of this type are announced, and furthermore they are put into context by reviewing from previous literature explicit product formulae for Jackson integrals relating to other roots systems obtained from the same perspective.

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Masahiko Ito. Peter J. Forrester. "Ramanujan’s $_{1}\psi_{1}$ summation theorem —perspective, announcement of bilateral $q$-Dixon–Anderson and $q$-Selberg integral extensions, and context—." Proc. Japan Acad. Ser. A Math. Sci. 90 (7) 92 - 97, July 2014. https://doi.org/10.3792/pjaa.90.92

Information

Published: July 2014
First available in Project Euclid: 7 August 2014

zbMATH: 1308.33015
MathSciNet: MR3249831
Digital Object Identifier: 10.3792/pjaa.90.92

Subjects:
Primary: 33D15 , 33D67
Secondary: 39A13

Keywords: Bailey’s very-well-poised $_{6}\psi_{6}$ summation formula , Dixon–Anderson integral , Ramanujan’s $_{1}\psi_{1}$ summation formula , Selberg Integral

Rights: Copyright © 2014 The Japan Academy

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Vol.90 • No. 7 • July 2014
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