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.

中文翻译：

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具有Liouvillian多双峰参数的Euler型序列的算术性质

摘要

本文指出，对于任何非零线性形式\（{{h} _ {0}} {{f} _ {0}}（1）+ {{h} _ {1}} {{f} _ {1 }}（1）\）具有整数系数h ₀，h ₁，存在无限多个p -adic场，这种形式不会消失。在这里，\（{{f} _ {0}}（1）= \ mathop \ sum \ limits_ {n = 0} ^ \ infty {{\ left（\ lambda \ right）} _ {n}} \）和\（{{f} _ {1}} \ left（1 \ right）= \ mathop \ sum \ limits_ {n = 0} ^ \ infty {{\ left（{\ lambda + 1} \ right）} _ { n}} \），其中λ是一个Liouvillian多阿迪数，（λ）_n代表Pochhammer符号。该结果表明研究具有先验参数的超几何级数的值的算术性质的可能性。