Extensions of two $q$-Gosper identities with an extra integer parameter

TL;DR

This paper extends two $q$-Gosper identities by adding an extra integer parameter, generalizing Gosper's $_3F_2$-series identities, and explores related results using series rearrangement methods.

Contribution

It introduces novel extensions of $q$-Gosper identities with an additional parameter, broadening the scope of known hypergeometric series identities.

Findings

Extended two $q$-Gosper identities with an extra parameter

Derived generalizations of Gosper's $_3F_2$-series identities

Presented several related hypergeometric series results

Abstract

According to the method of series rearrangement, we establish the extensions of two $q$-Gosper identities with an extra integer parameter. The limiting cases of them produce the generalizations of Gosper's two $_3F_2(\frac{3}{4})$-series identities with an additional integer parameter. Meanwhile, several related results are also given.