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Asymmetric Rogers--Ramanujan type identities. I. The Andrews--Uncu Conjecture (2203.15168v1)
Published 29 Mar 2022 in math.NT
Abstract: In this work, we start an investigation of asymmetric Rogers--Ramanujan type identities. The first object is the following unexpected relation $$\sum_{n\ge 0} \frac{(-1)n q{3\binom{n}{2}+4n}(q;q3)_n}{(q9;q9)_n} = \frac{(q{4};q{6})_\infty (q{12};q{18})\infty}{(q{5};q{6})\infty (q{9};q{18})_\infty}$$ 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.