Two closed-form evaluations for the generalized hypergeometric function ${}_4F_3(\frac1{16})$

Arjun K. Rathie, Mykola A. Shpot

Published 2023-03-14, updated 2024-08-12Version 2

The objective of this short note is to provide two closed-form evaluations for the generalized hypergeometric function $_4F_3$ of the argument $\frac1{16}$. This is achieved by means of separating a generalized hypergeometric function $_3F_2$ into even and odd components, together with the use of two known results for $_3F_2(\pm\frac14)$ available in the literature. As an application, we obtain an interesting infinite-sum representation for the number $\pi^2$. Certain connections with the work of Ramanujan and other authors are discussed, involving other special functions and binomial sums of different kinds.