The Sums of a Double Hypergeometric Series and of the First m+1 Terms of 3F2(a,b,c;(a+b+1)/2,2c;1) when c = -m is a Negative Integer

TL;DR

This paper derives a summation formula for the initial terms of a specific hypergeometric series when a parameter is a negative integer, enabling the evaluation of related double hypergeometric series sums.

Contribution

It introduces a new summation formula for the first m+1 terms of a particular 3F2 hypergeometric series with c = -m, facilitating the summation of related double hypergeometric series.

Findings

Derived a summation formula for 3F2 series with c = -m.

Applied the formula to evaluate a specific double hypergeometric series.

Provided a new tool for summing terminating hypergeometric series.

Abstract

A summation formula is derived for the sum of the first m+1 terms of the 3F2(a,b,c;(a+b+1)/2,2c;1) series when c = -m is a negative integer. This summation formula is used to derive a formula for the sum of a terminating double hypergeometric series that arose in another project by one of us (C.D.)