On the even-indexed eigenfunctions of the quantum harmonic oscillator

Abstract

In 2021, Fassari et al. introduced a remarkable summation involving the even-indexed eigenfunctions of the quantum harmonic oscillator, and introduced a proof, based on manipulations of multiple elliptic integrals, for an evaluation involving Catalan's constant for the aforementioned summation. This motivates the development of generalizations and extensions of this evaluation. In this article, we show, using the Wilf–Zeilberger method, how this series evaluation may be extended to infinite families of hypergeometric expressions that have free parameters and that are explicitly evaluable in terms of the digamma function. We also consider how an identity related to a nonterminating q-Pfaff–Saalschütz sum due to Krattenthaler and Srivastava may be applied to further generalize the main result due to Fassari et al.