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 ${}2\psi{2}$, we derive three type translation formulas for the generalized Zwegers' $\mu$-function (``continuous $q$-Hermite function'') which was introduced by Shibukawa--Tsuchimi (SIGMA, 2023). From some Bailey's transformation formula of ${}2\psi{2}$, we also give a formula for the expression of the generalized Zwegers' $\mu$-function by some very-well-poised bilateral basic hypergeometric series ${}4\psi{8}$. As an application of this new expression formula for the generalized Zwegers' $\mu$-function, we obtain some new Fourier expansions for the Weierstrass and Jacobi elliptic functions.