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arXiv · 2206.15137

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

Genki Shibukawa·Satoshi Tsuchimi
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Abstract

We introduce a one parameter deformation of the Zwegers' $μ$-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 $μ$-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 $μ$-function, and the Zwegers' $μ$-function is regarded as a continuous $q$-Hermite polynomial of ''$-1$ degree''.

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Genki Shibukawa, Satoshi Tsuchimi. 2023-03-23. A Generalization of Zwegers' $μ$-Function According to the $q$-Hermite-Weber Difference Equation. https://doi.org/10.3842/sigma.2023.014

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