On simultaneous approximations

P. L. Ivankov

Čebyševskij sbornik, Tome 16 (2015) no. 3, pp. 295-305

Résumé

In this paper we consider hypergeometric functions and their derivatives (see (2) and (3)). One begins the investigation of arithmetic nature of the values of such functions with the construction of functional linear approximating form having sufficiently high order of zero at the origin. If the parameters of functions under consideration (in our case these are numbers (1)) are rational the construction of such a form can be fulfilled by means of the Dirichlet principle. Further reasoning is based on the employment of the constructed form and the whole scheme is called Siegel's method, see [1] and [2]. If some of the numbers (1) are irrational the functions (2) and (3) cannot be reduced to the so called $E$-functions and it is impossible to use Siegel's method (in its classic form) for such functions: the scheme doesn't work at the very beginning of reasoning for we cannot use the Dirichlet principle for the construction of the first approximating linear functional form (in the process of reasoning by Siegel's method we get several such forms). It was noticed that in some cases the first approximating form can be constructed effectively (see for example [3] and [4]). Having at one's disposal such a form one can reason as in Siegel's method (or it is possible in some cases to use special properties of the effectively constructed linear form) and receive required results. These results are not so general as those received by Siegel's method but the method based on effectively constructed approximating form has its own advantages. One of them consists in the possibility of its application also in case when some of the parameters (1) are irrational. The other advantage is the more precise estimates (if we consider for instance the measure of linear independence) that can be obtained by this method. The above concerns the case when the functions under consideration are not differentiated with respect to parameter. Application of Siegel's method for the differentiated with respect to parameter functions (for example such functions as (4) and (5)) is possible also and it has been in fact fulfilled in a series of works; see the remarks to chapter 7 of the book by A. B. Shidlovskii [5]. But as before the parameters of the functions under consideration must be rational and the obtained results are not sufficiently precise. The performed investigations show that the employment of simultaneous approximations instead of construction of linear approximating form almost always gives better results. For that reason the main new results concerning differentiated with respect to parameter hypergeometric functions have been obtained exactly by means of the effective constructions of simultaneous approximations although the appearance (comparatively recently) of effective constructions of linear approximating forms for such functions did make it possible to solve some related problems. In this paper we propose a new effective construction of simultaneous approximations for the differentiated with respect to parameter hypergeometric functions in homogeneous case. On possible applications of this construction we give only brief instructions: one can obtain some results on linear independence of the values of functions of the type (5) in case of irrationality of some of the numbers (1); it is possible also to improve some of the related quantitative results. Bibliography: 15 titles.

Détail

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