Ivankov

pp. 1–12

pp. 102-113

pp,65-74

To study the arithmetic properties of values of the generalized hypergeometric functions with irrational parameters it is impossible to use directly Siegel's method known in the theory of tran-scendental numbers. The reason is a too fast growing minimal common denominator of the ex-pansion coefficients of such functions in formal power series. In some cases, however, this diffi-culty can be overcome by means of special reasoning, and a linear approximating form (or sim-ultaneous approximations) can be constructed using a Dirichlet principle. Here, some techniques related to the effective construction methods of the abovementioned approximations are applied too. In case of inhomogeneous forms, using these considerations leads to insufficiently accurate estimates. In this paper for refining such estimates we construct simultaneous approximations with the optimal choice of zero polynomial degree.

It is impossible to use the known in the theory of transcendental numbers Siegel's method directly for investigation of arithmetic properties of values of generalized hyper-geometric functions with irrational parameters because these functions do not belong to the class of E-functions. For that reason in such a situation one usually applies different variants of effective construction of linear approximating forms. In this paper we use one of such variants that makes it possible to consider the differentiated with respect to a parameter functions. The capabilities of this method are extended by a special choice of parameters of the functions under consideration.

In order to obtain quantitative results in the theory of Diophantine approximations one uses functional linear approximating forms which have sufficiently high order of zero at z = 0. Such forms are constructed either by means of the Dirichlet principle or effectively. In this article, by means of effective construction of approximating linear forms, we obtain a low estimate of the modulus of a linear form in the values of an exponential function in different points of an imaginary quadratic field; numerators of these points are roots of unity. Precise estimates, with respect to the height, were obtained with computation of corresponding constants. The proposed construction could be used for obtaining analogous estimates in values of generalized hypergeometric functions.

To study the arithmetic nature of the values ​​of hypergeometric functions with irrational parameters, effective Padé approximants of first or second kind are commonly used. In this case, the approximant of second kind (joint approach) are simpler and often produce more general results. In this paper the author proposes an effective construction of simultaneous approximations for hypergeometric functions of general form and their derivatives (including the parameter). With this construction, the corresponding module is estimated below by a linear form. Some of the parameters of the functions are irrational.

Investigation of arithmetic nature of values of generalized hypergeometric functions usually starts with applying a linear approximating form. Such a form must have a high order of zero at the coordinate basic origin and can be constructed by means of Dirichlet principle. The results obtained in this way are considerably general but the capabilities of such a method are restricted when precise quantitative assessment is required. Additional difficulties arise for functions with irrational parameters. In some cases the approximating form can be constructed effectively. Such a construction makes it possible to obtain more precise low estimates of linear forms for functions with rational parameters and to consider a case of irrational parameters. In this paper a new construction of Pade approximation for hypergeometric functions and their derivatives (also with respect to parameter) is proposed. This construction is applied for investigation of arithmetic properties of such functions.