ON DIFFERENTIATION WITH RESPECT TO PARAMETER
The investigation of the arithmetic nature of the values of diﬀerentiated with respect to parameter generalizedhypergeometric functions was carried out in many works; see [1]-[7] and also corresponding chapters of the books [8] and [9]. Primarily the method of Siegel was used for these purposes. This method canbe applied for the investigation ofhypergeometric functions with rational parameters and the results concerning transcendence and algebraic independence of the values of such functions and corresponding quantitative results (for example estimatesof the measuresof algebraic independence)were obtainedby means of it. Thepossibilities of application of Siegel’s method in case ofhypergeometric functions with irrational parameters are restricted. In its classic form Siegel’s method cannot be applied in this situation and here were required some new considerations. But it must be noted that the most general results concerningthe arithmetic natureofthevaluesofhypergeometric functions with irrational parameters were obtained exactly by Siegel’s method (by modiﬁed form of it, see [10] and [11]). In this case it’s impossible to say of the results of transcendence or algebraic independence and one must restrict oneself by the results concerning linear independence of the corresponding values. In Siegel’s method reasoning begins with the construction of functional linear approximating form which has a suﬃciently high order of zero at the originof coordinates. Such a formis constructedby meansof the Dirichlet principle. The impossibility to realize the corresponding reasoning for the functions with irrational parameters is an obstacle for the attempts to apply Siegel’s method in case of irrational parameters. It was noted long ago that in some cases the linear approximating form canbe constructed eﬀectively and explicit formulae canbepointed out for its coeﬃcients.Thismethodis inferiortoSiegel’soneinthe senseofthe generality of the results obtained. But by means of the method based on the eﬀective construction of linear approximating form the most precise low estimates of the modules of linear forms in the values of hypergeometric functions were obtained and in many cases were established linear independence of the values of functions with irrational parameters (see for example [12]). The eﬀective construction of linear approximating form for the function (2) was proposed in the work [13]. In this work the construction was based on a contour integral which was earlier used for the achievement of results concerning the estimatesof linear formsof thevaluesofhypergeometric functions with diﬀerent parameters; see [14]. In this paper we propose a new approach for the construction of linear approximating form for functions (2). Herewe make useofa connectionbetweenhypergeometric functionsof diﬀerenttypeswhichmakesitpossibleto reduceabovementioned constructingof linear approximating form to less diﬃcult task. In the conclusion we give short directions concerningpossible applications.