ON DIFFERENTIATION WITH RESPECT TO PARAMETER

The investigation of the arithmetic nature of the values of differentiated with respect to parameter generalized hypergeometric 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 can be applied for the investigation of hypergeometric functions with rational parameters and the results concerning transcendence and algebraic independence of the values of such functions and corresponding quantitative results (for example estimates of the measures of algebraic independence) were obtained by means of it. The possibilities of application of Siegel’s method in case of hypergeometric 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 concerning the arithmetic nature of the values of hypergeometric 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 origin of coordinates. Such a form is constructed by means of 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 can be constructed effectively and explicit formulae can be pointed out for its coefficients. This method is inferior to Siegel’s one in the sense of the 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 estimates of linear forms of the values of hypergeometric functions with different parameters; see [14]. In this paper we propose a new approach for the construction of linear approximating form for functions (2). Here we make use of a connection between hypergeometric functions of different types which makes it possible to reduce above mentioned constructing of linear approximating form to less difficult task. In the conclusion we give short directions concerning possible applications.

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2. Belogrivov, I. I. 1970, "The transcendence and algebraic independence of the values of certain hypergeometric E-functions" , Mat. Sb. (N.S.), vol. 82(124), pp. 387–408. (Russian)

3. Belogrivov, I. I. 1973, "The transcendence and algebraic independence of the values of a certain class of E-functions" , Sibirsk. Mat. Zh., vol. 14, pp. 16–35, 235. (Russian)

4. V¨a¨an¨anen, K. 1972, "On a cojecture of Mahler concerning the algebraic independence of the values of some E-functions" , Ann. Acad. Sci. Fennicae. Ser. A. Math., vol. 512, pp. 3–46.

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6. V¨a¨an¨anen, K. 1975, "On the algebraic independence of some E-functions related to Kummer’s functions" , Ann. Acad. Sci. Fennicae. Ser. A. Math., vol. 1, pp. 183–194.

7. V¨a¨an¨anen, K. 1975, "On the algebraic independence of the values of some Efunctions" , Ann. Acad. Sci. Fennicae. Ser. A. Math., vol. 1, pp. 93–109.

8. Mahler, K. 1976, "Lectures on Transcendental Numbers." Berlin: Springer Verlag.

9. Shidlovskii, A. B. 1987, "Transtsendentnye chisla" [Transcendental numbers] “Nauka”, Moscow, 448 pp. (Russian)

10. Galochkin, A. I. 1986, "An analogue of Siegel’s method" , Vestnik Moskov. Univ. Ser. I Mat. Mekh., no. 2, pp. 30–34, 113. (Russian)

11. Ivankov, P. L. 1993, "Linear independence of values of entire hypergeometric functions with irrational parameters" Sibirsk. Mat. Zh., vol. 34, no. 5, pp. 53– 62, ii, vii (Russian); translation in Siberian Math. J., vol. 34 (1993), no. 5, pp. 839—847.

12. Galochkin, A. I. 1984, "Estimates, unimprovable with respect to height, for certain linear forms" , Mat. Sb. (N.S.), vol. 124(166), no. 3, pp. 416–430. (Russian)

13. Ivankov, P. L. 2010, "On the differentiation of a hypergeometric function with respect to a parameter" , Fundam. Prikl. Mat. vol. 16, no. 6, pp. 91–94 (Russian); translation in J. Math. Sci. (N. Y.)vol. 182 (2012), no. 4, pp. 505–507.

14. Ivankov, P. L. 1992, "Arithmetic properties of values of hypergeometric functions with different parameters" , Mat. Zametki, vol. 52, no. 6, pp. 25–31, 157 (Russian) ; translation in Math. Notes, vol. 52 (1992), no. pp. 5–6, 1188—1192 (1993).

15. Ivankov, P. L. 1995, "On the linear independence of the values of some functions" , Fundam. Prikl. Mat., vol. 1, no. 1, pp. 191—206. (Russian)