The generalized Zwegers' $\mu$-function and transformation formulas for the bilateral basic hypergeometric series

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.