Proofs and reductions of various conjectured partition identities of Kanade and Russell

TL;DR

This paper proves seven conjectured Rogers-Ramanujan type identities modulo 12 by Kanade and Russell, including original and symmetric cases, and reduces four other conjectures to basic hypergeometric series.

Contribution

It provides rigorous proofs for seven conjectured identities and simplifies four additional conjectures to hypergeometric series forms.

Findings

Proved seven identities conjectured by Kanade and Russell.

Included original and symmetric identities related to affine Lie algebra modules.

Reduced four conjectures to single-sum basic hypergeometric series.

Abstract

We prove seven of the Rogers-Ramanujan type identities modulo $12$ that were conjectured by Kanade and Russell. Included among these seven are the two original modulo $12$ identities, in which the products have asymmetric congruence conditions, as well as the three symmetric identities related to the principally specialized characters of certain level $2$ modules of $A_9^{(2)}$. We also give reductions of four other conjectures in terms of single-sum basic hypergeometric series.