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arXiv · 1308.6650

The $q$-Dixon--Anderson integral and multi-dimensional $_1\psi_1$ summations

Masahiko Ito·Peter J. Forrester
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Abstract

The Dixon--Anderson integral is a multi-dimensional integral evaluation fundamental to the theory of the Selberg integral. The $_1\psi_1$ summation is a bilateral generalization of the $q$-binomial theorem. It is shown that a $q$-generalization of the Dixon--Anderson integral, due to Evans, and multi-dimensional generalizations of the $_1\psi_1$ summation, due to Milne and Gustafson, can be viewed as having a common origin in the theory of $q$-difference equations as expounded by Aomoto. Each is shown to be determined by a $q$-difference equation of rank one, and a certain asymptotic behavior. In calculating the latter, essential use is made of the concepts of truncation, regularization and connection formulae.

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Masahiko Ito, Peter J. Forrester. 2013-08-30. The $q$-Dixon--Anderson integral and multi-dimensional $_1\psi_1$ summations. https://arxiv.org/abs/1308.6650

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