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

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''.