Polyadic Liouville numbers
We study here polyadic Liouville numbers, which are involved in a series of recent papers. The author considered the series 𝑓0(𝜆) = ∞Σ 𝑛=0 (𝜆)𝑛𝜆𝑛, 𝑓1(𝜆) = ∞Σ 𝑛=0 (𝜆 + 1)𝑛𝜆𝑛, where 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 ≥ 1 we have(𝛾)𝑛 = 𝛾(𝛾 + 1)...(𝛾 + - 1). The values of these series were also calculated at polyadic Liouville number. The canonic expansion of a polyadic number is of the form = ∞Σ 𝑛=0 𝑎𝑛𝑛!, ∈ Z, 0 ≤ ≤ 𝑛. This series converges in any field of 𝑝-adic numbers Q𝑝. We call a polyadic number a polyadic Liouville number, if for any and there exists a positive integer such that for all primes ,satisfying ≤ 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.