Evaluations of series of the $q$-Watson, $q$-Dixon, and $q$-Whipple type

TL;DR

This paper extends classical $q$-series identities, specifically the $q$-Watson, $q$-Dixon, and $q$-Whipple formulas, by introducing additional parameters and deriving new generalizations using series rearrangement and transformation techniques.

Contribution

The paper introduces new extended formulas for $q$-series identities with extra parameters, expanding the scope of existing $q$-Watson, $q$-Dixon, and $q$-Whipple formulas.

Findings

Derived several extended $q$-Watson formulas with two extra parameters.

Generalized $q$-Dixon and $q$-Whipple formulas using Sears' transformation.

Presented numerous special series evaluations as particular cases.

Abstract

Using $q$-series identities and series rearrangement, we establish several extensions of $q$-Watson formulas with two extra integer parameters. Then they and Sears' transformation formula are utilized to derive some generalizations of $q$-Dixon formulas and $q$-Whipple formulas with two extra integer parameters. As special cases of these results, many interesting evaluations of series of $q$-Watson,$q$-Dixon, and $q$-Whipple type are displayed.