On the values of hypergeometric function with parameter from algebraic field of the fourth degree

Abstract

In order to investigate arithmetic properties of the values of generalized hypergeometric functions with rational parameters one often makes use of Siegel’s method. By means of this method have been achieved the most general results concerning this problem. The
main deficiency of Siegel’s method consists in the impossibility of its application in case of hypergeometric functions with irrational parameters. In this situation the investigation is usually based on the effective construction of the functional approximating form (in Siegel’s method the existence of such a form is proved by means of pigeon-hole principle). The construction and investigation of an approximating form is the first step to the achievement of arithmetic result.
Applying effective method we encounter at least two problems which make considerably narrow the area of its employment. First, the more or less general effective construction of the approximating form for the products of hypergeometric functions is unknown. While using
Siegel’s method one doesn’t deal with such a problem. Hence the investigator is compelled to consider only questions of linear independence of the values of hypergeometric functions over some algebraic field. Choosing this field is the second problem. The great majority of published results concerning corresponding questions deals with imaginary quadratic field (or the field of rational numbers). Only in exceptional situations it is possible to investigate the case of some other algebraic field. We consider here the case of a field of the fourth degree. By means of a special technique we
establish linear independence over such a field of the values of some hypergeometric function with irrational parameter from that field.