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

TL;DR

This paper generalizes Zwegers' $d$-function by introducing a one-parameter deformation linked to the $q$-Hermite-Weber equation, providing new formulas and relations, and connecting it to continuous $q$-Hermite polynomials.

Contribution

It presents a novel one-parameter deformation of Zwegers' $d$-function using $q$-Borel and $q$-Laplace transforms, expanding its analytical properties and connections to $q$-orthogonal polynomials.

Findings

Derived formulas for the generalized $d$-function, including shift, translation, and symmetry relations.

Established a difference equation for the new parameter of the $d$-function.

Connected the generalized $d$-function to continuous $q$-Hermite polynomials.

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''.