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arXiv · 2305.08101

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

Genki Shibukawa·Satoshi Tsuchimi
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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.

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Genki Shibukawa, Satoshi Tsuchimi. 2023-05-14. The generalized Zwegers' $\mu$-function and transformation formulas for the bilateral basic hypergeometric series. https://arxiv.org/abs/2305.08101

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