The Generalized Zwegers' μ-Function and Transformation Formulas for the Bilateral Basic Hypergeometric Series

Abstract

By applying Slater's transformation formulas for the bilateral basic hypergeometric series ₂ψ₂, we derive three translation or connection formulas for the generalized Zwegers' μ-function ("continuous q-Hermite function") which was introduced by Shibukawa–Tsuchimi (SIGMA, 2023). From some Bailey's transformation formula of ₂ψ₂, we also give a formula for the expression of the generalized Zwegers' μ-function by a certain degenerate Very-Well-Poised bilateral basic hypergeometric series ₄ψ₈. As an application of this new expression formula for the generalized Zwegers' μ-function, we obtain some new q-expansions for elliptic functions and Ramanujan's mock theta functions.