Extensions of the classical theorems for very well-poised hypergeometric functions

TL;DR

This paper extends classical theorems for very well-poised hypergeometric functions using Bailey's transform and related methods, leading to new summations, transformations, and applications in hypergeometric theory.

Contribution

It introduces novel extensions of key hypergeometric theorems by applying Bailey's transform and classical extension techniques, broadening the scope of hypergeometric identities.

Findings

Extended $_{5}F_4(1)$ summation theorem.

Derived new $_{7}F_6(1)$ and $_{9}F_8(1)$ transformations.

Generated new hypergeometric summations and transformations.

Abstract

The classical summation and transformation theorems for very well-poised hypergeometric functions, namely, $_{5}F_4(1)$ summation, Dougall's $_{7}F_6(1)$ summation, Whipple's $_{7}F_6(1)$ to $_{4}F_3(1)$ transformation and Bailey's $_{9}F_8(1)$ to $_{9}F_8(1)$ transformation are extended. These extensions are derived by applying the well-known Bailey's transform method along with the classical very well-poised summation and transformation theorems for very well-poised hypergeometric functions and the Rakha and Rathie's extension of the Saalsch\"{u}tz's theorem. To show importance and applications of the discovered extensions, a number of special cases are pointed out, which leads not only to the extensions of other classical theorems for very well-poised and well-poised hypergeometric functions but also generate new hypergeometric summations and transformations.