A Generalization of Zwegers' -Function According to the -Hermite-Weber Difference Equation - Shibukawa, Genki; Tsuchimi, Satoshi (Symmetry, Integrability and Geometry: Methods and Applications, 2023)
We introduce a one-parameter deformation of the Zwegers' -function as the image of -Borel and -Laplace transformations of a fundamental solution for the -Hermite-Weber equation. We further give some formulas for our generalized -function, for example, forward and backward shift, translation, symmetry, a difference equation for the new parameter, and bilateral -hypergeometric expressions. From one point of view, the continuous -Hermite polynomials are some special cases of our -function, and the Zwegers' -function is regarded as a continuous -Hermite polynomial of ''−1 degree''.