Arithmetic properties of values at polyadic Liouville points of Euler-type series with polyadic Liouville parameter

V. G. Chirskii

Čebyševskij sbornik, Tome 22 (2021) no. 2, pp. 304-312

Résumé

We study infinite linear independence of polyadic numbers $$ f_{0}(\lambda)=\sum_{n=0}^\infty (\lambda)_{n}\lambda^{n}, f_{1}(\lambda)=\sum_{n=0}^\infty (\lambda +1)_{n}\lambda^{n},$$ where $ \lambda $ is a certain polyadic Liouville number. The series considered converge in any field $ \mathbb{\mathrm{Q}}_p $. Here $(\gamma)_{n}$ denotes Pochhammer symbol, i.e. $(\gamma)_{0}=1$ , and for $n\geq 1$ we have$ (\gamma)_{n}=\gamma(\gamma+1)\dots(\gamma+n-1)$. The result extends the previous author's result on the polyadic numbers $$ f_{0}(1)=\sum_{n=0}^\infty (\lambda)_{n}, f_{1}(1)=\sum_{n=0}^\infty (\lambda +1)_{n}.$$ The values of generalized hypergeometric series are the subject of numerous studies. If the parameters of the series are rational numbers, then they come either in the class of $E$ (if these series are entire functions) or the class of $G$ functions (if they have a finite non-zero radius of convergence) or to the class of $F-$ series ( in the case of zero radius of convergence in the field of complex numbers, however, they converge in the fields of $p-$ adic numbers). In all these cases, the Siegel-Shidlovsky method and its generalizations are applicable. If the parameters of the series contain algebraic irrational numbers, then the study of their arithmetic properties is based on the Hermite-Pade approximations. In this case, the parameter is a transcendental number. It should be noted that earlier A. I. Galochkin proved the algebraic independence of the values of $E - $functions at a point that is a real Liouville number. We also mention the published works of E. Yu. Yudenkova on the values of $F - $ series in polyadic Liouville points. We especially note that in this paper we consider the values in the polyadic transcendental point of hypergeometric series, the parameter of which is the polyadic transcendental (Liouville) number. Note that earlier A.I. Galochkin proved the algebraic independence of values of $E-$functions at points which are real Liouville numbers. We also mention submitted papers (E.Yu.Yudenkova) about the arithmetic properties of values of $F-$series at polyadic Liouville numbers. It should be specially mentioned that here we study the values of hypergeometric series with a parameter which is a polyadic Liouville number.

Détail

Keywords: polyadic Liouville number, infinite linear independence.