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A Generalization of Zwegers' $\mu$-Function According to the $q$-Hermite–Weber Difference Equation
Genki Shibukawa, Satoshi Tsuchimi Department of Mathematics, Kobe University, Rokko, 657-8501, Japan
Аннотация:
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".
Ключевые слова:
Appell–Lerch series, $q$-Boerl transformation, $q$-Laplace transformation, $q$-hypergeometric series, continuous $q$-Hermite polynomial, mock theta functions.
Поступила: 2 июля 2022 г.; в окончательном варианте 25 февраля 2023 г.; опубликована 23 марта 2023 г.
Образец цитирования:
Genki Shibukawa, Satoshi Tsuchimi, “A Generalization of Zwegers' $\mu$-Function According to the $q$-Hermite–Weber Difference Equation”, SIGMA, 19 (2023), 014, 23 pp.
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/sigma1909 https://www.mathnet.ru/rus/sigma/v19/p14
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