Multiparameter Ramanujan-type \(q\)-beta integrals (Q1273311)

The integrals of the title are of the form \[ \int^\infty_{-\infty} \varrho(s) [x(s+\textstyle{{1\over 2}})- x(s- \textstyle{{1\over 2}})] ds,\tag{\(*\)} \] where \(\varrho\) is a solution of a Pearson-type equation and \(x(s)= C_1q^{-s}+ C_2q^s\). Slater has given a transformation formula expressing a bilateral \(q\)-series \({_r\psi_r}(z)\) as a linear combination of \(r\) such series. To this transformation, the author establishes analogues expressing integrals of the form \((*)\) with \(2r\) parameters in terms of \(r\) integrals over \(\mathbb{R}\). The very-well-poised case is separately considered. Next, proceeding in a slightly different manner, the author obtains transformations expressing, instead, the integrals in terms of \({_r\psi_r}\)-functions, and integrals of periodic functions from 0 and 1. These main results involve a number of free parameters, and for small values of \(r\) known theorems for \({_r\psi_r}\) can be utilized, such that by specialization some known and some new \(q\)-beta integrals emerge.