The Clausenian hypergeometric function $_3F_2$ with unit argument and negative integral parameter differences

TL;DR

This paper derives new explicit and symmetric summation formulas for the Clausenian hypergeometric series $_3F_2(1)$ with negative integral parameter differences, extending previous results and highlighting connections to earlier work.

Contribution

It introduces novel explicit and symmetric three-term summation formulas for $_3F_2(1)$ with negative parameter differences, generalizing prior relations.

Findings

Derived new explicit summation formulas

Established symmetric three-term relations

Connected results to historical work by P. W. Karlsson

Abstract

New explicit as well as manifestly symmetric three-term summationformulas are derived for the Clausenian hypergeometric series $_3F_2(1)$ with negative integral parameter differences. Our results generalize and naturally extend several similar relations published, in recent years, by many authors. An appropriate and useful connection is established with the quite underestimated 1974 paper by P. W. Karlsson.