Geodesic
A Generalization of Zwegers' $\mu$-Function According to the $q$-Hermite–Weber Difference Equation
Symmetry, integrability and geometry: methods and applications, Tome 19 (2023) Cet article a éte moissonné depuis la source Math-Net.Ru

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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".
Keywords: $q$-hypergeometric series, continuous $q$-Hermite polynomial, mock theta functions.
Mots-clés : Appell–Lerch series, $q$-Boerl transformation, $q$-Laplace transformation 
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     author = {Genki Shibukawa and Satoshi Tsuchimi},
     title = {A {Generalization} of {Zwegers'} $\mu${-Function} {According} to the $q${-Hermite{\textendash}Weber} {Difference} {Equation}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2023},
     volume = {19},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a13/}
}
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Genki Shibukawa; Satoshi Tsuchimi. A Generalization of Zwegers' $\mu$-Function According to the $q$-Hermite–Weber Difference Equation. Symmetry, integrability and geometry: methods and applications, Tome 19 (2023). http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a13/

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