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Chebyshevskii Sbornik, 2021, Volume 22, Issue 3, Pages 245–255
DOI: https://doi.org/10.22405/2226-8383-2018-22-3-245-255
(Mi cheb1073)
 

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)
Full-text PDF (695 kB) Citations (2)
References:
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
English version:
Doklady Mathematics (Supplementary issues), 2022, Volume 106, Issue 2, Pages 137–141
DOI: https://doi.org/10.1134/S1064562422700314
Document Type: Article
UDC: 511.36
Language: Russian
Citation: V. G. Chirskii, “Polyadic Liouville numbers”, Chebyshevskii Sb., 22:3 (2021), 245–255; Doklady Mathematics (Supplementary issues), 106:2 (2022), 137–141
Citation in format AMSBIB
\Bibitem{Chi21}
\by V.~G.~Chirskii
\paper Polyadic Liouville numbers
\jour Chebyshevskii Sb.
\yr 2021
\vol 22
\issue 3
\pages 245--255
\mathnet{http://mi.mathnet.ru/cheb1073}
\crossref{https://doi.org/10.22405/2226-8383-2018-22-3-245-255}
\transl
\jour Doklady Mathematics (Supplementary issues)
\yr 2022
\vol 106
\issue 2
\pages 137--141
\crossref{https://doi.org/10.1134/S1064562422700314}
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  • https://www.mathnet.ru/eng/cheb/v22/i3/p245
  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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