Arithmetic Properties of Generalized Hypergeometric F-Series

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 Kv\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{K}_v$$\end{document}-completions of an algebraic number fieldK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{K}$$\end{document} of finite degree over the field of rational numbers with respect to a valuation v of the field K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{K}$$\end{document} extending the p-adic valuation of the field ℚ over all primes p except for finitely many of them.