Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 014, 23 pages      arXiv:2206.15137      https://doi.org/10.3842/SIGMA.2023.014

A Generalization of Zwegers' $\mu$-Function According to the $q$-Hermite-Weber Difference Equation

Genki Shibukawa and Satoshi Tsuchimi
Department of Mathematics, Kobe University, Rokko, 657-8501, Japan

Received July 02, 2022, in final form February 25, 2023; Published online March 23, 2023

Abstract
We introduce a one parameter deformation of the Zwegers' $\mu$-function as the image of $q$-Borel and $q$-Laplace transformations of a fundamental solution for the $q$-Hermite-Weber equation. We further give some formulas for our generalized $\mu$-function, for example, forward and backward shift, translation, symmetry, a difference equation for the new parameter, and bilateral $q$-hypergeometric expressions. From one point of view, the continuous $q$-Hermite polynomials are some special cases of our $\mu$-function, and the Zwegers' $\mu$-function is regarded as a continuous $q$-Hermite polynomial of ''$-1$ degree''.

Key words: Appell-Lerch series; $q$-Boerl transformation; $q$-Laplace transformation; $q$-hypergeometric series; continuous $q$-Hermite polynomial; mock theta functions.

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