Three new $q$-Abel transformations and their applications

TL;DR

This paper introduces three novel $q$-Abel transformation formulas for $q$-series, derived using Abel's lemma, and applies them to establish new $q$-contiguous relations and transformations.

Contribution

The paper presents three new $q$-Abel transformations and their applications to $q$-series, expanding the toolkit for $q$-series transformations and relations.

Findings

Three new $q$-Abel transformation formulas established.

Derived $q$-contiguous relations for truncated $q$-series.

Several new $q$-series transformations obtained.

Abstract

In the present paper, we establish three special $q$-Abel transformation formulae of $q$-series via the use of Abel's lemma on summation by parts. As direct applications, we set up the corresponding $q$-contiguous relations for three kinds of truncated $q$-series. Several new transformations are consequently established.