On linear approximating forms

P. L. Ivankov

Čebyševskij sbornik, Tome 21 (2020) no. 1, pp. 200-212

Résumé

Generalized hypergeometric function is defined as a sum of the power series whose coefficients are the products of the values of some fractional rational function. Taken with a minus sign roots of a numerator and denominator of this rational function are called parameters of the corresponding hypergeometric function. For the investigation of the arithmetic nature of the values of hypergeometric functions and their derivatives (including derivatives with respect to parameter) one often makes use of Siegel's method. The corresponding reasoning begins as a rule by the construction of the functional linear approximating form. If parameters of the hypergeometric function are rational one is able to use pigeonhole principle for the construction of this form. In addition the construction is feasible not only for the hypergeometric functions themselves but also for the products of their powers. By this is explained the generality of results obtained by such method. But if there are irrational numbers among the parameters the application of a pigeonhole method is impossible and for carrying out the corresponding investigation it is necessary to employ some additional considerations. One of the methods of surmounting the difficulty connected with the irrationality of some parameters of a hypergeometric function consists in the application of the effective construction of the linear approximating form from which the reasoning begins. Primarily effective constructions of such approximations appeared for the functions of a special kind (the numerator of the rational function by means of which the coefficients of hypergeometric functions are defined was to be equal to unity). The investigation of the properties of these approximations revealed the fact that they can be useful in case of rational parameters as well for the quantitative results obtained by effective methods turned out to be more precise than their analogs obtained by Siegel's method. Subsequently the methods of effective construction of linear approximating forms were generalized in diverse directions. In this paper we propose a new effective construction of approximating form in case when for the hypergeometric functions derivatives with respect to parameter are also considered. This construction is made use of for the sharpening of the lower estimates of the linear independence measure of the values of corresponding functions.

Détail

Keywords: hypergeometric functions, linear independence, differentiation with respect to parameter, estimates of linear forms.