Proofs of some partition identities conjectured by Kanade and Russell

TL;DR

This paper provides new proofs for several Rogers-Ramanujan-type partition identities conjectured by Kanade and Russell, using quadratic transformations of special polynomials, and includes some previously unproven cases and related identities.

Contribution

It introduces novel proof techniques for conjectured partition identities, expanding the set of identities with rigorous demonstrations and connecting them to special polynomial transformations.

Findings

New proofs for five conjectures by Kanade and Russell.

Proofs of four previously open conjectures.

A new proof of Capparelli's partition identity.

Abstract

Kanade and Russell conjectured several Rogers-Ramanujan-type partition identities, some of which are related to level $2$ characters of the affine Lie algebra $A_9^{(2)}$. Many of these conjectures have been proved by Bringmann, Jennings-Shaffer and Mahlburg. We give new proofs of five conjectures first proved by those authors, as well as four others that have been open until now. Our proofs for the new cases use quadratic transformations for Askey-Wilson and Rogers polynomials. We also obtain some related results, including a new proof of a partition identity conjectured by Capparelli and first proved by Andrews.