On Some Hypergeometric Summations

TL;DR

This paper investigates non-terminating hypergeometric sums with a focus on conditions under which they can be expressed as gamma product formulas, using complex analysis, geometry, and number theory techniques.

Contribution

It introduces new necessary arithmetic conditions for hypergeometric sums to have gamma product formulas, extending classical results in hypergeometric series.

Findings

Derived necessary conditions for gamma product formulas in hypergeometric sums

Applied complex and asymptotic analysis to transcendental curves

Explored rational approximations of irrational numbers in this context

Abstract

We develop a theoretical study of non-terminating hypergeometric summations with one free parameter. Composing various methods in complex and asymptotic analysis, geometry and arithmetic of certain transcendental curves and rational approximations of irrational numbers, we are able to obtain some necessary conditions of arithmetic flavor for a given hypergeometric sum to admit a gamma product formula. This kind of research seems to be new even in the most classical case of the Gauss hypergeometric series.