Integrals and Series Representations of $q$-Polynomials and Functions: Part II Schur Polynomials and the Rogers-Ramanujan Identities

TL;DR

This paper explores new integral and series representations of $q$-polynomials, focusing on Schur polynomials and Rogers-Ramanujan identities, providing extensions and special cases that deepen understanding of basic hypergeometric functions.

Contribution

It introduces novel expansions and identities involving the Ramanujan function $A_q$ and Stieltjes--Wigert polynomials, including $m$-versions of Rogers-Ramanujan type identities.

Findings

Derived new identities involving $A_q$ and Stieltjes--Wigert polynomials.

Established $m$-versions of Rogers-Ramanujan identities.

Explored bilateral extensions of transformations in basic hypergeometric functions.

Abstract

We give several expansion and identities involving the Ramanujan function $A_q$ and the Stieltjes--Wigert polynomials. Special values of our idenitities give $m$-versions of some of the items on the Slater list of Rogers-Ramanujan type identities. We also study some bilateral extensions of certain transformations in the theory of basic hypergeometric functions.