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

Abstract

Bhoria, Eyyunni and Maji recently obtained a four-parameterq-series identity whichgives as special cases not only all five entries of Ramanujan on pages 354 and 355 of his secondnotebook 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 naturallygives finite analogues of Ramanujan’s re-sults. Using one of these finite analogues, we deduce an identity for a finite sum involving a 2φ1. Thisidentity is then applied to obtain a generalization of the generating function version of Andrews’ fa-mous identity for the smallest parts function spt(n). Theq-series which generalizes ∑∞n=1spt(n)qn is completely different fromS(z,q) considered by Andrews, Garvan and Liang. Further applications ofour identity are given. Lastly we generalize a result of Andrews, Chan and Kim which involves the firstodd moments of rank and crank.