Arithmetic Properties of an Euler-Type Series with Polyadic Liouville Parameter

Abstract

Series of the form

$$\sum_{n=0}^\infty (\lambda)_{n}z^{n},\sum_{n=0}^\infty (\lambda+1)_{n}z^{n}$$

are studied. By definition, the Pochhammer symbol $(\gamma)_{n}$ is equal to $(\gamma)_{0}=1$ and $(\gamma)_{n}=\gamma(\gamma+1)\cdots(\gamma+n-1)$ for $n\geq 1$. These series resemble the series $\sum_{n=0}^{\infty}n!(-z)^{n}$ known to Euler, which belong to generalized hypergeometric series.

The polyadic numbers are the direct product of rings of $p-$adic integers. Polyadic numbers have a canonical representation in the form of a series $\sum_{n=0}^{\infty}a_{n}n!$, where $a_{n}$ are integers satisfying $0\leq a_{n}\leq n.$ This series converges in any field $\mathbb{\mathrm{Q}}_p$, and its sum in this field is an integer $p-$adic number. If, for any positive integers $n$ and $P$, there is a positive integer $A$ such that the inequality $\left|\lambda -A \right|_{p}< A^{-n}$ holds for all primes $p$, $p\leq P$, then the polyadic number $\lambda$ is called a Liouville polyadic number. In this inequality, $\lambda$ denotes the sum of the series that specifies the polyadic number $\lambda$in the field $\mathbb{\mathrm{Q}}_p.$ It can readily be seen that any Liouville polyadic number is a transcendental element of any field of $p-$adic numbers. It is important that, unlike previous works, here $\lambda$ is a polyadic Liouville number.

Let us formulate the main result of the paper.

Let $\lambda_0=0$ and let $\mu_0$ be an arbitrary positive integer. Write $\lambda_1= \lambda_0 + \mu_0$ and $s_1= \exp(\lambda_1).$ Let $\mu_1$ be an arbitrary positive integer such that the inequality $ord_p\mu_1\geq 2s_1 \ln s_1$ holds for any prime $p\leq s_1+\lambda_{1}+1$. For $k\geq2$ we write $\lambda_k = \lambda_{k-1} + \mu_{k-1}, s_k = \exp\lambda_k$; let a positive integer $\mu_k$ be chosen in such a way that, for $p \leq s_k + \lambda_{k}+1$, we have the inequality $ord_p\mu_k\geq 2s_k \ln s_k.$ Let

$$\lambda = \sum_{k=0}^\infty \mu_k.$$

The main result of the paper is the proof of the following theorem.

Theorem. For any integers $h_{0}$ and $h_{1}$ that are not equal to zero simultaneously, there exists an infinite set of primes $p$ such that the inequality

$$|L|_{p}=|h_{0}f_{0}(1)+h_{1}f_{1}(1)|_{p}>0$$

holds in the field $\mathbb{\mathrm{Q}}_p$.