On Differentiation with Respect to Parameter of a Hypergeometric Function of a Special Type

Abstract

We propose an effective construction of Padé approximation for the hypergeometric function of special type and its derivative with respect to parameter and this parameter is included both in numerator and denominator of the general term of the corresponding series. The aforementioned construction has been realized by means of a modification of the method that was applied earlier in simpler situations. Thus obtained approximation is made use of afterwards to establish the lower estimates of the numerical linear forms in the values of the functions under consideration. One could obtain similar estimates by the known in the theory of transcendental numbers Siegel’s method, however the estimates obtained by this method would be of less exactness.