Polyadic Liouville numbers

V. G. Chirskii

Čebyševskij sbornik, Tome 22 (2021) no. 3, pp. 245-255.

Résumé

We study here polyadic Liouville numbers, which are involved in a series of recent papers. The author considered the series $$ f_{0}(\lambda)=\sum_{n=0}^\infty (\lambda)_{n}\lambda^{n}, f_{1}(\lambda)=\sum_{n=0}^\infty (\lambda +1)_{n}\lambda^{n},$$ where $ \lambda $ is a certain polyadic Liouville number. The series considered converge in any field $ \mathbb{\mathrm{Q}}_p $. Here $(\gamma)_{n}$ denotes Pochhammer symbol, i.e. $(\gamma)_{0}=1$, and for $n\geq 1$ we have$ (\gamma)_{n}=\gamma(\gamma+1)\dots(\gamma+n-1)$. The values of these series were also calculated at polyadic Liouville number. The canonic expansion of a polyadic number $\lambda$ is of the form $$ \lambda= \sum_{n=0}^\infty a_{n} n!, a_{n}\in\mathbb{\mathrm{Z}}, 0\leq a_{n}\leq n.$$ This series converges in any field of $p$-adic numbers $ \mathbb{\mathrm{Q}}_p $. We call a polyadic number $\lambda$ a polyadic Liouville number, if for any $n$ and $P$ there exists a positive integer $A$ such that for all primes $p$, satisfying $p\leq P$ the inequality $$\left|\lambda -A \right|_{p}^{-n}$$ holds. The paper gives a simple proof that the Liouville polyadic number is transcendental in any field $\mathbb{\mathrm{Q}}_p.$ In other words,the Liouville polyadic number is globally transcendental. We prove here a theorem on approximations of a set of $p$-adic numbers and it's corollary — a sufficient condition of the algebraic independence of a set of $p$-adic numbers. We also present a theorem on global algebraic independence of polyadic numbers.

Détail

Keywords: polyadic number, polyadic Liouville number.

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