$q$-Bessel Functions and Rogers-Ramanujan Type Identities

TL;DR

This paper explores $q$-Bessel functions, generalizes Ramanujan functions, extends Rogers-Ramanujan identities, and derives new generating functions and identities related to these special functions.

Contribution

It introduces a generalized Ramanujan function, extends Rogers-Ramanujan identities, and provides new generating functions for Stieltjes-Wigert polynomials.

Findings

Evaluation of $q$-Bessel functions at infinite points

Extension of $m$-version of Rogers-Ramanujan identities

New Rogers-Ramanujan type identities

Abstract

We evaluate $q$-Bessel functions at an infinite sequence of points and introduce a generalization of the Ramanujan function and give an extension of the $m$-version of the Rogers-Ramanujan identities. We also prove several generating functions for Stieltjes-Wigert polynomials with argument depending on the degree. In addition we give several Rogers-Ramanujan type identities.