Polyadic Liouville numbers

Abstract

We study here polyadic Liouville numbers, which are involved in a series of recent papers. The author considered the series
𝑓0(𝜆) =∞Σ︁𝑛=0(𝜆)𝑛𝜆𝑛, 𝑓1(𝜆) =∞Σ︁𝑛=0(𝜆 + 1)𝑛𝜆𝑛,
where 𝜆 is a certain polyadic Liouville number. The series considered converge in any field Q𝑝 . Here (𝛾)𝑛 denotes Pochhammer symbol, i.e. (𝛾)0 = 1 , and for 𝑛 ≥ 1 we have(𝛾)𝑛 =
𝛾(𝛾 + 1)...(𝛾 + 𝑛 − 1). The values of these series were also calculated at polyadic Liouville number. 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.
The paper gives a simple proof that the Liouville polyadic number is transcendental in any field Q𝑝. 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.