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By applying Slater's transformation formulas for the bilateral basichypergeometric series ${}_2\psi_{2}$, we derive three type translation formulasfor the generalized Zwegers' $\mu$-function (``continuous $q$-Hermitefunction'') which was introduced by Shibukawa--Tsuchimi (SIGMA, 2023). Fromsome Bailey's transformation formula of ${}_2\psi_{2}$, we also give a formulafor the expression of the generalized Zwegers' $\mu$-function by somevery-well-poised bilateral basic hypergeometric series ${}_4\psi_{8}$. As anapplication of this new expression formula for the generalized Zwegers'$\mu$-function, we obtain some new Fourier expansions for the Weierstrass andJacobi elliptic functions.

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