  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.
