Extensions of classical hypergeometric identities of Bailey and Whipple

TL;DR

This paper extends classical hypergeometric identities, providing new transformation formulas for various series types, and demonstrates how these generalizations encompass many well-known results in the field.

Contribution

It introduces a method to derive higher-order hypergeometric transformations from lower-order series and extends classical quadratic transformations by taking limits.

Findings

New transformations of nearly-poised and very-well-poised series.

Extensions of classical identities of Bailey and Whipple.

Connections to well-known hypergeometric results.

Abstract

We obtain extensions of classical hypergeometric identities of Bailey and Whipple that transform nearly-poised and very-well-poised series to Saalsch\"utzian series, Saalsch\"utzian series to Saalsch\"utzian series, and very-well-poised and nearly-poised series to very-well-poised series. We employ a method in which summations and transformations of lower-order series are used to obtain transformations of higher-order series. By taking limits, we also obtain extensions of two classical quadratic transformations of Whipple and Bailey. Furthermore, we show how a number of other well-known results regarding hypergeometric series follow as special cases of our results.