Values of hypergeometric $F$-series at polyadic Liouvillea points

E. Yu. Yudenkova

Čebyševskij sbornik, Tome 22 (2021) no. 2, pp. 536-542

Résumé

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: $$ \sum_{n=0}^{\infty} \frac{\left(\alpha_{1}\right)_{n} \cdots\left(\alpha_{r}\right)_{n}}{\left(\beta_{1}\right)_{n} \ldots\left(\beta_{s}\right)_{n} n !} z^{n}. $$ $F$ – series have form $f_n = \sum_{n=0}^{\infty}a_n n! z^n$ whose coefficients $a_n$ satisfy some arithmetic properties. These series converge in the field $\mathbb{Q}_p$ of $p$ – adic numbers and their algebraic extensions $\mathbb{K}_v$. Polyadic number is a series of the form $\sum_{n=0}^{\infty} a_nn!, a_n \in \mathbb{Z}$. 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 \left| x - \frac{p}{q} \right| \frac{1}{q^n}. $ The polyadic Liouville number $\alpha$ has the property that for any numbers $P, D$ there exists an integer $|A|$ such that for all primes $p \leq P$ the inequality $|\alpha - A|_p A^{-D}. $

Détail

Keywords: hypergeometric $F$-series, polyadic Liouville numbers.