On Polyadic Liouville numbers

Abstract

We study here polyadic Liouville numbers, which are involved in a series of recent papers. The canonic expansion of a polyadic number 𝜆 is of the form

$$𝜆 =∞Σ︁𝑛=0𝑎𝑛𝑛!, 𝑎𝑛 ∈ Z, 0 ≤ 𝑎𝑛 ≤ 𝑛.$$

This series converges in any field of 𝑝− adic numbers Q𝑝 .
We call a polyadic number 𝜆 a polyadic Liouville number, if for any 𝑛 and 𝑃 there exists a positive integer 𝐴 such that for all primes 𝑝 ,satisfying 𝑝 ≤ 𝑃 the inequality

$$|𝜆 − 𝐴|𝑝 < 𝐴^(−𝑛)$$

holds.
Let 𝑘 ≥ 2 be a positive integer. We denote for a positive integer 𝑚

$$Φ(𝑘,𝑚) = 𝑘^𝑘^(...)^𝑘$$

Let

$$𝑛𝑚 = Φ(𝑘,𝑚)$$

and let

$$𝛼 =∞Σ︁𝑚=0(𝑛𝑚)!.$$

Theorem 1. For any positive integer 𝑘 ≥ 2 and any prime number 𝑝 the series 𝛼 converges to a transcendental element of the ring Z𝑝. In other words, the polyadic number 𝛼 is globally transcendental.