Double-sum Rogers-Ramanujan type identities

TL;DR

This paper derives new Rogers-Ramanujan type identities involving double sums by expanding Chebyshev polynomials in terms of $q$-Hermite polynomials and utilizing their orthogonality, generalizing previous results.

Contribution

It introduces a novel method to generate Rogers-Ramanujan identities using $q$-Hermite polynomial expansions and orthogonality relations.

Findings

Derived new Rogers-Ramanujan type identities involving double sums.

Generalized previous identities by Andrews, Shi, Sun, and Yao.

Abstract

As the $q$-analog of Chebyshev polynomials, $q$-Hermite polynomials form a cornerstone in the family of $q$-orthogonal polynomials, which play a fundamental role in quantum algebra and mathematical physics. Recently, Andrews obtained a series of Rogers-Ramanujan type identities by constructing Bailey pairs from Chebyshev polynomials. In this paper, by applying the expansion formula of Chebyshev polynomials in terms of $q$-Hermite polynomials and using the orthogonality relations, we derive a series of Rogers-Ramanujan type identities on double sums, which further generalized the known results due to Andrews, Shi, Sun and Yao.