Barnes and Ramanujan-Type Integrals on the q-Linear Lattice

Abstract

Let C be a contour in the complex s-plane and $\rho (s)$ be the solution of a Pearson-type first-order difference equation with coefficient functions $\sigma (s)$ and $\tau (s)$, on the q-linear lattice $x(s) = q^{ - s} $, $0 < q < 1$. For the cases in which (i) $\sigma (s)$, $\tau (s)$ are polynomials of degrees 2 and 1, respectively, and (ii) $\sigma (s)$ is a polynomial of degree 2 but $\tau (s)$ has a simple pole, the integral $\int_C {\rho (s)} q^{ - s} ds$ is considered. When C is the whole real line, q-analogues of some formulas due to Ramanujan are obtained, and some of the questions raised in a previous paper are resolved. When C is along the imaginary axis, the iteration technique in the parameters of $\rho (s)$ works, permitting alternative proofs of a formula due to Askey and Roy, as well as its extension by Gasper, to be given.