Arithmetic Properties of Euler-Type Series with a Liouvillian Polyadic Parameter

Abstract

This paper states that, for any nonzero linear form ${{h}_{0}}{{f}_{0}}(1) + {{h}_{1}}{{f}_{1}}(1)$ with integer coefficients h₀, h₁, there exist infinitely many p-adic fields where this form does not vanish. Here, ${{f}_{0}}(1) = \mathop \sum \limits_{n = 0}^\infty {{\left( \lambda \right)}_{n}}$ and ${{f}_{1}}\left( 1 \right) = \mathop \sum \limits_{n = 0}^\infty {{\left( {\lambda + 1} \right)}_{n}}$, where λ is a Liouvillian polyadic number and (λ)_n stands for the Pochhammer symbol. This result shows the possibility of studying the arithmetic properties of values of hypergeometric series with transcendental parameters.