The q-Dixon–Anderson integral and multi-dimensional ψ11 summations

https://doi.org/10.1016/j.jmaa.2014.10.034Get rights and content
Under an Elsevier user license
open archive

Abstract

The Dixon–Anderson integral is a multi-dimensional integral evaluation fundamental to the theory of the Selberg integral. The ψ11 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 ψ11 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 1, and a certain asymptotic behavior. In calculating the latter, essential use is made of the concepts of truncation, regularization and connection formulae.

Keywords

Dixon–Anderson integral
Evans's q-integral
Ramanujan's ψ11 summation formula
Milne–Gustafson sum
Gustafson's An sum
Aomoto method

Cited by (0)