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

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.