New Proofs of Some Double Sum Rogers-Ramanujan Type Identities

TL;DR

This paper introduces new proofs for certain double sum Rogers-Ramanujan type identities using an integral method, simplifying previous combinatorial and $q$-difference approaches, and confirms a conjecture by Andrews and Uncu.

Contribution

The paper provides streamlined integral-based proofs for double sum Rogers-Ramanujan identities and verifies a conjecture by Andrews and Uncu, expanding the toolkit for such identities.

Findings

New proofs for double sum Rogers-Ramanujan identities

Relation of double sums to known single sum identities

Confirmation of Andrews and Uncu's conjectural identity

Abstract

Recently, Rosengren utilized an integral method to prove a number of conjectural identities found by Kanade and Russell. Using this integral method, we give new proofs to some double sum identities of Rogers-Ramanujan type. These identities were earlier proved by approaches such as combinatorial arguments or by using $q$-difference equations. Our proofs are based on streamlined calculations, which relate these double sum identities to some known Rogers-Ramanujan type identities with single sums. Moreover, we prove a conjectural identity of Andrews and Uncu which was earlier confirmed by Chern.