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

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.