Transcendence of certain $2$-adic numbers

V. G. Chirskii

Čebyševskij sbornik, Tome 24 (2023) no. 5, pp. 237-243.

Résumé

We prove here that at least one of the two 2-adic numbers which are the values at $z=1$ of the sums in $ \mathbb{\mathrm{Q}}_2 $ of 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 $ .We deal here with $ \mathbb{\mathrm{Q}}_2 $. The symbol $(\gamma)_{n}$ denotes Pochhammer symbol, i.e. $(\gamma)_{0}=1$ , and for $n\geq 1$ we have$ (\gamma)_{n}=\gamma(\gamma+1)...(\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 $$\left|\lambda -A \right|_{p}|A|^{-n}$$ holds. It was proved earlier that the Liouville polyadic number is transcendental in any field $\mathbb{\mathrm{Q}}_p.$ In other words,the Liouville polyadic number is globally transcendental. It allowed to prove using some equality that there exists an infinite set of $p-$adic fields $ \mathbb{\mathrm{Q}}_p $ where at least one of the numbers $f_{0}(z),f_{1}(z).$ Here we prove the transcendence of values in the field $ \mathbb{\mathrm{Q}}_2 $.

Détail

Keywords: transcendence, polyadic number, polyadic Liouville number,.

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