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.