О значениях гипергеометрических функций

Abstract

The investigation of arithmetic properties of the values of the generalized hypergeometric
functions is often carried out by means of known in the theory of transcendental numbers Siegel’s
method. The most general results in this field have been obtained precisely by this method.
But the possibilities of Siegel’s method in case of hypergeometric functions with irrational
parameters are restricted. This is connected with the fact that such hypergeometric functions are
not E-functions and for that reason one is unable to construct linear approximating form with
large order of zero by means of pigeonhole method. To consider problems connected with the
investigation of arithmetic properties of the values of hypergeometric functions with irrational
parameters it is possible in some cases to use the method based on the effective construction
of linear approximating form but the possibilities of this method are also limited because of
the absence of too general effective constructions. There are some difficulties also in the cases
when such constructions are available. The peculiarities of these constructions often hinder the
realization of arithmetic part of the method.
For that reason of some interest are situations when one is able to realize the required
investigation by means of specific properties of concrete functions. Sometimes it is possible to
choose the parameters of the functions under consideration in such a way that one receives
the possibility to overcome the difficulties of the general case. In this paper we consider
hypergeometric function of a special kind and its derivatives. By means of effective construction
it is possible not only to prove linear independence of the values of this function and its
derivatives over some imaginary quadratic field but also to obtain corresponding quantitative
result in the form of the estimation the modulus of the linear form in the aforesaid values.