General $q$-series transformations based on Abel's lemma on summation by parts and their applications

TL;DR

This paper develops new general transformations for q-series using Abel's lemma, leading to novel hypergeometric series transformations and unifying various multibasic transformations.

Contribution

It introduces three new transformations with multiple parameters based on Abel's lemma, expanding the toolkit for q-series transformations and unifying existing results.

Findings

New q-series transformations with sixteen parameters

Derived new quadratic, cubic, and quartic hypergeometric transformations

Unified multibasic transformations through (R,S)-type framework

Abstract

In this paper, we establish three new and general transformations with sixteen parameters and bases via Abel's lemma on summation by parts. As applications, we set up a lot of new transformations of basic hypergeometric series. Among include some new results of Gasper and Rahman's quadratic, cubic, and quartic transformations. Furthermore, we put forward the so-called $(R,S)$-type transformation with arbitrary degree to unify such multibasic transformations. Some special $(R,S)$-type transformations are presented.