On Polyadic Liouville Numbers

The study of polyadic Liouville numbers has begun relatively recently. The canonical expansion of a polyadic number λ is of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda = \sum\limits_{n = 0}^\infty {{a}_{n}}n!,{{a}_{n}} \in {\text{Z}},0 \leqslant {{a}_{n}} \leqslant n.$$\end{document} This series converges in any field Qp of p-adic numbers. A polyadic number λ is called a polyadic Liouville number (or a Liouville polyadic number) if for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document} and P there exists a positive integer A such that for all primes p satisfying \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \leqslant P$$\end{document} the inequality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\left| {\lambda - A} \right|}_{p}} < {{A}^{{ - n}}}$$\end{document} holds. Given a positive integer m, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi (k,m) = {{k}^{{{{k}^{{{{{...}}^{k}}}}}}}}$$\end{document} denote the result of k raised to the power k successively m times. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{n}_{m}} = \Phi (k,m)$$\end{document}, and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = \sum\limits_{m = 0}^\infty ({{n}_{m}})!.$$\end{document} Theorem 1 states that, for any positive integer \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \geqslant 2$$\end{document} and any prime number p, the series \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} converges to a transcendental element of the ring Zp. In other words, the polyadic number α is globally transcendental.

n n n n a n a a n n ≤ p P 1. INTRODUCTION This work is concerned with the theory of transcendental numbers in non-Archimedean valuation domains and is a direct continuation of [1].
Recall that the ring of polyadic integers is defined as the direct product of rings of p-adic integers over all primes p. Thus, elements of this ring can be treated as infinite-dimensional vectors, whose coordinates in the corresponding ring of p-adic integers are denoted by . As a result, we can introduce the concepts of algebraic, transcendental, infinitely transcendental, and globally transcendental polyadic numbers and the n n n n a n a a n λ λ ( ) p concepts of algebraic dependence, algebraic independence, infinite algebraic independence, and global algebraic independence of a set of polyadic numbers (see [9]). A polyadic number is called a polyadic Liouville number (or a Liouville polyadic number) if for any numbers and P there exists a positive integer A such that for all primes p satisfying the inequality it is true that More precisely, we should write but, by convention, when considering a field of p-adic numbers, by the symbol we mean the sum of this series.

MAIN RESULTS
Let be a positive integer. Let , and, for a positive integer m, let denote the result of k raised to the power k successively m times. In other words, Let us prove that for it holds that (6) where is a function tending to as Definitions (2) and (5) imply that Additionally, combining (1), (2), and (4) obviously yields the inequality (7) It follows from (1) that (8) (which holds as equality for k = 2). We use the wellknown relation where M is an arbitrary positive integer and is the sum of the digits in the p-adic expansion of M. It follows that (9) Thus, fixing k, we obtain (10) To prove inequality (6), in view of (7) and (10), it suffices to prove that This inequality is equivalent to Using the obvious inequality we conclude that (11) follows from the inequality (12) Consider the function It increases for Therefore, if (13) then, according to (8), inequality (12) follows from (14) Since as inequalities (13) and (14) hold for if the function is defined, for example, as It remains to apply Theorem 1 from [1], which is stated below.

Theorem [1, Theorem 1]. Any polyadic Liouville number is transcendental in any field
In other words, any polyadic Liouville number is globally transcendental.
Let m be a positive integer. Define (15) where the symbol denotes the integer part of the number . For any positive integer N, let N = lm + i and, for each let Additionally, let (17) (18) Theorem 2. The polyadic numbers α i , , ..., m -1 are globally algebraically independent.
The proof of this theorem relies heavily on the following result.
Theorem [1,Theorem 3]. For all , let where as As in (7) and (10), taking into account (24) and (25), we have Therefore, inequality (26) follows from In view of (9), it follows from the inequality (27) In turn, inequality (27) follows from the rough estimate (28) By definition (15), and the definition of the function immediately implies that as . Therefore, condition (28) obviously holds with the corresponding function . The verification of the asymptotic equalities (23) reduces to checking the relations which obviously hold as follows from definition (15).