Ramanujan’s $_{1}\psi_{1}$ summation theorem —perspective, announcement of bilateral $q$-Dixon–Anderson and $q$-Selberg integral extensions, and context—

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.