On linear approximating forms

Abstract

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.