Arithmetic Properties of Generalized Hypergeometric F-Series

Abstract

In the paper, using a generalization of the Siegel-Shidlovskii method in the theory of transcendental numbers, we prove the infinite algebraic independence of elements, generated by generalized hypergeometric series, of direct products of the fields of $$\mathbb{K}_v$$-completions of an algebraic number field$$\mathbb{K}$$ of finite degree over the field of rational numbers with respect to a valuation v of the field $$\mathbb{K}$$ extending the p-adic valuation of the field ℚ over all primes p except for finitely many of them.