Asymmetric Rogers–Ramanujan type identities. I. The Andrews–Uncu conjecture

Abstract

In this work, we start an investigation of asymmetric Rogers–Ramanujan type identities. The first object is the following unexpected relation ∑ n ≥ 0 ( − 1 ) n q 3 ( n 2 ) + 4 n ( q ; q 3 ) n ( q 9 ; q 9 ) n = ( q 4 ; q 6 ) ∞ ( q 12 ; q 18 ) ∞ ( q 5 ; q 6 ) ∞ ( q 9 ; q 18 ) ∞ \begin{equation*} \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 } \end{equation*} and its a a -generalization. We then use this identity as a key ingredient to confirm a recent conjecture of G. E. Andrews and A. K. Uncu [Ramanujan J. (2023), DOI 10.1007/s11139-022-00685-y].