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This article is cited in 2 scientific papers (total in 2 papers)
Polyadic Liouville numbers
V. G. Chirskiiab a Lomonosov Moscow State University (Moscow)
b RANEPA (Moscow)
Abstract:
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 $$\left|\lambda -A \right|_{p}<A^{-n}$$ 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.
Keywords:
polyadic number, polyadic Liouville number.
Received: 11.06.2021 Accepted: 20.09.2021
Citation:
V. G. Chirskii, “Polyadic Liouville numbers”, Chebyshevskii Sb., 22:3 (2021), 245–255
Linking options:
https://www.mathnet.ru/eng/cheb1073 https://www.mathnet.ru/eng/cheb/v22/i3/p245
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Abstract page: | 116 | Full-text PDF : | 41 | References: | 18 |
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