Asymptotic formulae of two divergent bilateral basic hypergeometric series

Abstract

We provide new formulae for the degenerations of the bilateral basic hypergeometric function ${}_1\psi_1 ( a; b; q, z )$ with using the $q$-Borel-Laplace transformation. These are thought of as the first step to construct connection formulae of $q$-difference equation for ${}_1\psi_1 ( a; b; q, z )$. Moreover, we show that our formulae have the $q \to 1 - 0$ limit.