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- Title
Polyadic Liouville Numbers.
- Authors
Chirskii, V. G.
- Abstract
The study of polyadic Liouville numbers has begun relatively recently. They make up an important part of the author's works concerning the infinite linear independence of the polyadic numbers where is a polyadic Liouville number. Here, denotes the Pochhammer symbol, i.e., and, for , The considered series converge in any field Qp. A parameter of the considered Euler-type series is a polyadic Liouville number, and the values of these series are calculated at a polyadic Liouville point. We note E.S. Krupitsyn's works establishing estimates for polynomials in sets of polyadic Liouville numbers and Yudenkova's works in which the values of F-series are considered at polyadic Liouville points. The canonic expansion of a polyadic number is of the form This series converges in any field Qp of p-adic numbers. A polyadic number is called a polyadic Liouville number (or a Liouville polyadic number) if for any n and P there exists a positive integer A such that for all primes p satisfying the inequality holds. We prove a simple statement that a Liouville polyadic number is transcendental in any field In other words, a Liouville polyadic number is globally transcendental. Additionally, a theorem is proved about the properties of approximations of a set of p-adic numbers, and its corollary is established, which is a sufficient condition for the algebraic independence of a set of p-adic numbers. A theorem on the global algebraic independence of polyadic numbers is obtained.
- Subjects
INTEGERS; POLYNOMIALS; P-adic analysis; LINEAR dependence (Mathematics)
- Publication
Doklady Mathematics, 2022, Vol 106, pS137
- ISSN
1064-5624
- Publication type
Article
- DOI
10.1134/S1064562422700314