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

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 quadratic, cubic, and quartic transformations. Furthermore, we put forward the so-called $(R,S)$-type transformation with arbitrary degree to unify such multibasic transformations. A special $(3,3)$-type transformation and its example are presented.

Keywords:

• q-series
• Abel's lemma
• algebraic identity
• indefinite summation
• transformation formula
• $(R,S)$-type

AMS Subject Classifications (2020):

• Primary 33D15
• Secondary 05A30

Acknowledgments

The authors thank the anonymous referees for their careful reading and insightful comments on the first version of this manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [Grant No. 11971341].