Two closed-form evaluations for the generalizedhypergeometric function \({}_{4}F_{3}\left(\frac{1}{16}\right)\)

Abstract

The objective of this short note is to provide two closed-form evaluations for the generalized hypergeometric function   \({}_4F_3\) of the argument \(\frac{1}{16}\) 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 \({}_3F_2(\pm\frac{1}{4})\) 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.