The $q$-Dixon--Anderson integral and multi-dimensional ${}_1{\psi }_1$ summations

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.