Some q-Identities derived by the ordinary derivative operator

TL;DR

This paper explores the use of the ordinary derivative operator in q-series theory, deriving new summation and transformation formulas related to classical identities like the q-binomial theorem and Rogers' identities.

Contribution

It introduces novel q-series identities and formulas by applying the ordinary derivative operator, offering alternative proofs and finite forms of well-known results.

Findings

New summation formulas related to q-series

Finite form of Rogers-Ramanujan identity

Simplified proof of Eisenstein's Lambert series theorem

Abstract

In this paper, we investigate applications of the ordinary derivative operator, instead of the $q$-derivative operator, to the theory of $q$-series. As main results, many new summation and transformation formulas are established which are closely related to some well-known formulas such as the $q$-binomial theorem, Ramanujan's ${}_1\psi_1$ formula, the quintuple product identity, Gasper's $q$-Clausen product formula, and Rogers' ${}_6\phi_5$ formula, etc. Among these results is a finite form of the Rogers-Ramanujan identity and a short way to Eisenstein's theorem on Lambert series.