Two closed-form evaluations for the generalized hypergeometric function ${}_4{F}_3(\frac{1}{16})$

Abstract

The objective of this short note is to provide two closed-form evaluations for the generalized hypergeometric function ${}_4{F}_3$ of the argument $\frac{1}{16}$. This is achieved by means of separating a generalized hypergeometric function ${}_3{F}_2$ into even and odd components, together with the use of two known results for ${}_3{F}_2(\pm \frac{1}{4})$ available in the literature. As an application, we obtain an interesting infinite-sum representation for the number $\ddot{I}{€}^2$. Certain connections with the work of Ramanujan and other authors are discussed, involving other special functions and binomial sums of different kinds.

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