Asymmetric Rogers--Ramanujan type identities. I. The Andrews--Uncu Conjecture

TL;DR

This paper explores asymmetric Rogers--Ramanujan identities, introduces a new unexpected relation, and confirms a recent conjecture by Andrews and Uncu using this identity as a key tool.

Contribution

It presents a novel asymmetric Rogers--Ramanujan type identity and verifies a conjecture by Andrews and Uncu with this new relation.

Findings

Discovered an unexpected Rogers--Ramanujan type identity.

Confirmed the Andrews--Uncu conjecture using the new identity.

Provided an $a$-generalization of the identity.

Abstract

In this work, we start an investigation of asymmetric Rogers--Ramanujan type identities. The first object is the following unexpected relation $$\sum_{n\ge 0} \frac{(-1)^n q^{3\binom{n}{2}+4n}(q;q^3)_n}{(q^9;q^9)_n} = \frac{(q^{4};q^{6})_\infty (q^{12};q^{18})_\infty}{(q^{5};q^{6})_\infty (q^{9};q^{18})_\infty}$$ and its $a$-generalization. We then use this identity as a key ingredient to confirm a recent conjecture of G. E. Andrews and A. K. Uncu.