Variations on a Hypergeometric Theme

TL;DR

This paper investigates the possibility of expressing a specific hypergeometric function in closed form, ultimately concluding it is unlikely, while also deriving interesting related identities and transformations for hypergeometric and sum functions.

Contribution

The paper introduces new identities and transformations for hypergeometric functions and generalized sums, expanding the mathematical literature.

Findings

The specific hypergeometric function cannot be expressed in closed form.

Derived new $_4F_3(1)$, $_5F_6(1)$, and Euler sum identities.

Presented educational methods and previously undocumented identities.

Abstract

The question was asked: Is it possible to express the function \begin{equation} \tag{1.1} h(a)\equiv\,{_4F_3}(a,a,a,a;2a,a+1,a+1;1) \label{question} \end{equation} in closed form? After considerable analysis, the answer appears to be "no", but during the attempt to answer this question, a number of interesting (and unexpected) related results were obtained, either as specialized transformations, or as closed-form expressions for several related functions. The purpose of this paper is to record and review both the methods attempted and the related identities obtained (specifically new $_4F_3(1)$, $_5F_6(1)$ and (generalized Euler) sums containing digamma functions) - the former for their educational merit, since they appear to be not-very-well-known, the latter because they do not appear to exist in the literature.