A finite analogue of a $q$-series identity of Bhoria, Eyyunni and Maji and its applications

TL;DR

This paper develops a finite analogue of a complex $q$-series identity, extending Ramanujan's results and applying it to generalize identities related to the smallest parts function and moments of rank and crank.

Contribution

It introduces a finite analogue of a $q$-series identity of Bhoria, Eyyunni, and Maji, enabling new finite sum identities and generalizations of classical partition function results.

Findings

Derived a finite analogue of a $q$-series identity.

Obtained a new identity for a finite sum involving ${}_2\phi_1$.

Generalized Andrews' identity for the smallest parts function.

Abstract

Bhoria, Eyyunni and Maji recently obtained a four-parameter $q$-series identity which gives as special cases not only all five entries of Ramanujan on pages 354 and 355 of his second notebook but also allows them to obtain an analytical proof of a result of Bressoud and Subbarao. Here, we obtain a finite analogue of their identity which naturally gives finite analogues of Ramanujan's results. Using one of these finite analogues, we deduce an identity for a finite sum involving a ${}_2\phi_1$. This identity is then applied to obtain a generalization of the generating function version of Andrews' famous identity for the smallest parts function $\textup{spt}(n)$. The $q$-series which generalizes $\sum_{n=1}^{\infty}\textup{spt}(n)q^n$ is completely different from $S(z, q)$ considered by Andrews, Garvan and Liang. Further applications of our identity are given. Lastly we generalize a result of Andrews, Chan and Kim which involves the first odd moments of rank and crank.