О значениях гипергеометрической функции с параметром из квадратичного поля

Abstract

In order to investigate arithmetic properties of the values of generalized hypergeometric\xa0functions with rational parameters one usually applies Siegel’s method. By means of this\xa0method have been achieved the most general results concerning the above mentioned properties. The main deficiency of Siegel’s method consists in the impossibility of its application for the\xa0hypergeometric functions with irrational parameters. In this situation the investigation is usually\xa0based on the effective construction of the functional approximating form (in Siegel’s method\xa0the existence of that form is proved by means of pigeon-hole principle). The construction and\xa0investigation of such a form is the first step in the complicated reasoning which leads to the achievement of arithmetic result.\xa0Applying effective method we encounter at least two problems which make extremely\xa0narrow the field of its employment. First, the more or less general effective construction of\xa0the approximating form for the products of hypergeometric functions is unknown. While using\xa0Siegel’s method one doesn’t deal with such a problem. Hence the investigator is compelled to\xa0consider only questions of linear independence of the values of hypergeometric functions over\xa0some algebraic field. Choosing this field is the second problem. The great majority of published\xa0results concerning corresponding questions deals with imaginary quadratic field (or the field of\xa0rational numbers). Only in exceptional situations it is possible to investigate the case of some\xa0other algebraic field. We consider here the case of a real quadratic field. By means of a special technique we\xa0establish linear independence of the values of some hypergeometric function with irrational\xa0parameter over such a field.