On Approximation of Hyper-geometric Function Values of a Special Class

Abstract

Investigations of arithmetic properties of the hyper-geometric function values make it possible to single out two trends, namely, Siegel’s method and methods based on the effective construction of a linear approximating form. There are also methods combining both approaches mentioned. The Siegel’s method allows obtaining the most general results concerning the abovementioned problems. In many cases it was used to establish the algebraic independence of the values of corresponding functions. Although the effective methods do not allow obtaining propositions of such generality they have nevertheless some advantages. Among these advantages one can distinguish at least two: a higher precision of the quantitative results obtained by effective methods and a possibility to study the hyper-geometric functions with irrational parameters. In this paper we apply the effective construction to estimate a measure of the linear independence of the hyper-geometric function values over the imaginary quadratic field. The functions themselves were chosen by a special way so that it could be possible to demonstrate a new approach to the effective construction of a linear approximating form. This approach makes it possible also to extend the well-known effective construction methods of the linear approximating forms for poly-logarithms to the functions of more general type. To obtain the arithmetic result we had to establish a linear independence of the functions under consideration over the field of rational functions. It is apparently impossible to apply directly known theorems containing sufficient (and in some cases needful and sufficient) conditions for the system of functions appearing in the theorems mentioned. For this reason, a special technique has been developed to solve this problem. The paper presents the obtained arithmetic results concerning the values of integral functions, but, with appropriate alterations, the theorems proved can be adapted to the case of the hyper-geometric series with a finite radius of convergence. Subsequently, it will be possible to consider also a case of the irrational parameters, but there are still a number of difficulties, which are specific for the problems of such type and which are to be overcome.