Values of hypergeometric 𝐹-series at polyadic Liouvillea points

Abstract

This paper proves infinite algebraic independence of the values of hypergeometric 𝐹 – series at polyadic Liouville points. Hypergeometric functions are defined for |𝑧| < 1 by the power series:

$$Σ︁𝑛=0((𝛼1)𝑛 · · · (𝛼𝑟)𝑛)/((𝛽1)𝑛 . . . (𝛽𝑠)𝑛 𝑛!)𝑧^𝑛$$.

𝐹 – series have form 𝑓𝑛 = Σ︀∞ 𝑛=0 𝑎_𝑛𝑛!𝑧^𝑛 whose coefficients 𝑎𝑛 satisfy some arithmetic properties.
These series converge in the field Q𝑝 of 𝑝 – adic numbers and their algebraic extensions K𝑣. Polyadic number is a series of the form Σ︀∞𝑛=0 𝑎_𝑛𝑛!, 𝑎_𝑛 ∈ Z. Liouville number is a real number x
with the property that, for every positive integer n, there exist infinitely many pairs of integers (𝑝, 𝑞) with 𝑞 > 1 such that 0 <|𝑥 − 𝑝/𝑞| < 1/𝑞^𝑛 . The polyadic Liouville number 𝛼 has the property that for any numbers 𝑃,𝐷 there exists an integer |𝐴| such that for all primes 𝑝 ≤ 𝑃 the inequality |𝛼 − 𝐴|𝑝 < 𝐴^(-𝐷).