On the values of hypergeometric functions

P. L. Ivankov

Čebyševskij sbornik, Tome 21 (2020) no. 2, pp. 159-168

Résumé

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.

Détail

Keywords: hypergeometric function, effective construction, linear independence, imaginary quadratic field.