Arithmetic Properties of Values at Polyadic Liouville Points of Euler-Type Series with Polyadic Liouville Parameter

We study the infinite linear independence of polyadic numbers , f 1 (λ) = where λ is a polyadic Liouville number. Here, denotes the Pochhammer symbol, i.e., and for . The considered series converge in any field . The results presented extends the author’s previous results concerning the arithmetic properties of the polyadic numbers . The values of generalized hypergeometric series have been extensively studied. If the series parameters are rational numbers, then the series belong to the class of E -functions (if these series are entire functions), to the class of G -functions (if they have a finite nonzero radius of convergence), or to the class of F -series (in the case of a zero radius of convergence in the field of complex numbers, but they converge in the fields of p -adic numbers). In all these cases, the Siegel–Shidlovskii 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 Hermite–Padé approximations. In the considered case, the parameter is a transcendental number. Note that earlier A.I. Galochkin proved the algebraic independence of the values of E -functions at points that are real Liouville numbers. We also mention E.Yu. Yudenkova’s papers (submitted for publication) on the values of F -series at polyadic Liouville points. It should be emphasized that this paper deals with the values, at polyadic transcendental points, of hypergeometric series with a parameter that is a polyadic transcendental (Liouville) number.