On Values Approximation of Some Special Type Hypergeometric Functions with Irrational Parameters

Abstract

In this paper we investigate arithmetic nature of the values of generalized hypergeometric functions and their derivatives. To solve the problem one often makes use of Siegel’s method. The first step in corresponding reasoning is, using the pigeonhole principle, to construct a functional linear approximating form, which has high order of zero at the origin of the coordinates.A hypergeometric function is defined as a sum of a power series whose coefficients are the products of the values of some rational function. Taken with the opposite sign, the zeroes of a numerator and a denominator of this rational function are called parameters of the corresponding generalized hypergeometric function. If the parameters are irrational it is impossible, as a rule, to employ Siegel’s method. In this case one applies the method based on the effective construction of the linear approximating form.Additional difficulties arise in case the rational function numerator involved in the formation of the coefficients of the hypergeometric function under consideration is different from the identical unit. In this situation even the availability of the effective construction of approximating form does not enable achieving an arithmetic result yet. In this paper we consider just such a case. To overcome difficulties arisen here we consider the values of the corresponding hypergeometric function and its derivatives at small points only and impose additional restrictions on parameters of the function.