On Some Hypergeometric Summations

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.