paper

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

arXiv:2203.15168

Abstract

In this work, we start an investigation of asymmetric Rogers--Ramanujan type identities. The first object is the following unexpected relation $n \geq 0 \sum \frac{( - 1 ) ^{n} q ^{3 (2 n) + 4 n} ( q ; q ^{3} ) _{n}}{( q ^{9} ; q ^{9} ) _{n}} = \frac{( q ^{4} ; q ^{6} ) _{\infty} ( q ^{12} ; q ^{18} ) _{\infty}}{( q ^{5} ; q ^{6} ) _{\infty} ( q ^{9} ; q ^{18} ) _{\infty}}$ 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.