One variable Generalization of five entries of Ramanujan and their finite analogue

TL;DR

This paper develops a one-variable generalization of Ramanujan's five q-series identities and their finite analogues, extending recent work and unifying several previous results in the theory of q-series.

Contribution

It introduces a novel one-variable generalization of Bhoria et al.'s identity and its finite analogue, which also encompasses Ramanujan's identities.

Findings

Derived a one-variable generalization of Bhoria et al.'s identity.

Established finite analogues that extend Dixit and Patel's results.

Unified several identities of Ramanujan through new generalizations.

Abstract

Ramanujan recorded five $q$-series identities at the end of his second notebook and an unified generalization of these identities obtained by Bhoria, Eyyunni and Maji. Recently, Dixit and Patel gave a finite analogue of the identity of Bhoria et. al. which in turn gives finite analogue of all the aforementioned identities of Ramanujan. In this paper, one of our main goals is to obtain a one-variable generalization of the identity of Bhoria et. al. along with its finite analogue, which naturally generalizes the result of Dixit and Patel. Utilizing these newly established identities, we derive one-variable generalizations for each of the five entries by Ramanujan and their corresponding finite analogues.