On the Hyper-geometric Function Value Approximation to the Parameter from the Real Quadratic Field

• Main Author: P. L. Ivankov
• Format: Article
• Language: Russian
• Published: MGTU im. N.È. Baumana 2017-04-01
• Series: Matematika i Matematičeskoe Modelirovanie
• Subjects: generalized hypergeometric functions; quadratic irrationality; estimate of linear form
• Online Access: https://www.mathmelpub.ru/jour/article/view/57

While studying arithmetic properties of the values of the generalized hyper-geometric functions there is always a need, arising in the process of reasoning, to have the lower estimate of the modulus of a nonzero algebraic integer. This estimate meets all the requirements only if the above-mentioned algebraic integer is rational or belongs to some imaginary quadratic field. It is by no means always possible to overcome difficulties caused by fact that a nonzero algebraic integer from the arbitrary algebraic field may be arbitrarily small.Additional problems arise from the fact that the least common denominator of the first  coefficients of the hyper-geometric series with irrational parameters grows too fast if  tends to infinity. The last circumstance makes it impossible to use a Dirichlet principle for the construction of the initial functional approximating form, and the construction of such a form is usually the first step on the way to obtain the corresponding arithmetic result.Because of two above-mentioned difficulties, numerous theorems concerning arithmetic properties of the sums of generalized hyper-geometric series with rational parameters cannot be extended to the case when the parameters are taken from the arbitrary field of the algebraic numbers.In this paper we consider a special type of hyper-geometric function the only parameter of which is a real quadratic irrationality. The above-mentioned difficulties have been overcome here in several steps. The linear approximating form from which a consideration begins is constructed by a special method that simultaneously uses the elements of two different approaches to such a construction: an application of the Dirichlet principle is combined with an effective method. This step is not carried out explicitly in the paper, since the earlier proved theorems are referred to. The difficulty due to the fact that the absolute value of an integer from a real quadratic field can be arbitrarily small has been overcome by means of a certain identity from a theory of special functions. We use also some special techniques to refine the corresponding quantitative results obtained earlier.