On approximation of the values of some hypergeometric functions with irrational parameters

P. L. Ivankov

Čebyševskij sbornik, Tome 17 (2016) no. 1, pp. 108-116.

Résumé

In this paper we consider some hypergeometric functions whose parameters are connected in a special way. Lower estimates of the moduli of linear forms in the values of such functions have been obtained. Usually for the achievement of such estimates one makes use of Siegel's method; see [1], [2], [3, chapt. 3]. In this method the reasoning begins with the construction by means of Dirichlet principle of the linear approximating form having a sufficiently large order of zero at the origin of coordinates. Employing the system of differential equations, the functions under consideration satisfy, one constructs then a set of forms such that the determinant composed of the coefficients of the forms belonging to this set must not be equal to zero identically. Further steps consist of constructing a set of numerical forms and of proving of the interesting for the researcher assertions: linear independence of the values of the functions under consideration can be proved or corresponding quantitative results can be obtained. By means of Siegel's method have been proved sufficiently general theorems concerning the arithmetic nature of the values of the generalized hypergeometric functions and in addition to aforementioned linear independence in many cases was established the transcendence and algebraic independence of the values of such functions. But the employment of Dirichlet principle at the first step of reasoning restricts the possibilities of the method. Its direct employment is possible in the case of hypergeometric functions with rational parameters only. It must be taken into consideration also the insufficient accuracy of the quantitative results that can be obtained by this method. As a consequence of these facts some analogue of Siegel's method has been developed (see [4]) by means of which it became possible in some cases to investigate the arithmetic nature of the values of hypergeometric functions with irrational parameters also. But yet earlier one had begun to apply methods based on effective construction of linear approximating form. By means of such constructions the arithmetic nature of some classic constants was investigated and corresponding quantitative results were obtained, see for example [5, chapt. 1]. Subsequently it turned out that effective methods can be applied also for the investigation of generalized hypergeometric functions. Explicit formulae for the coefficients of the linear approximating forms were obtained. In some cases these formulae make it possible to realize Siegel method scheme also for the hypergeometric functions with irrational parameters. If in (1) polynomial $a(x)$ is equal to unity identically then the results obtained by effective method are of sufficiently general nature and in this case further development of this method meets the obstacles of principal character. In case $a(x)\not\equiv1$, however, the possibilities of effective method are not yet exhausted and the latest results can be generalized and improved. In the theorems proved in the present paper new qualitative and quantitative results are obtained for some hypergeometric functions with $a(x)=x+\alpha$ and polynomial $b(x)$ from (1) of special character. The case of irrational parameters is under consideration but the ideas we use will apparently make it possible in the future to obtain new results in case of rational parameters also. Bibliography: 15 titles.

Détail

Keywords: generalized hypergeometric functions, irrational parameters, estimates of linear forms.

Bibliographie

[1] Siegel C. L., “Über einige Anwendungen Diophantischer Approximationen”, Abh. Preuss. Acad. Wiss., Phys.-Math. Kl., 1929–1930, no. 1, 1–70 | Zbl

[2] Siegel C. L., Transcendental numbers, Princeton University Press, Princeton, 1949 | MR | Zbl

[3] Shidlovskii A. B., Transcendental numbers, Nauka, M., 1987, 447 pp. (Russian) | MR

[4] Galochkin A. I., “An analogue of Siegel's method”, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (Moscow Univ. Math. Bulletin), 1986, no. 2, 30–34 (Russian)

[5] Fel'dman N. I., Hilbert's seventh problem, Moskov. Gos. Univ., M., 1982, 312 pp. (Russian)

[6] Ivankov P. L., “Lower estimates of the linear forms in the values of Kummer's function”, Mat. Zametki (Mathematical Notes), 49:2 (1991), 55–63 (Russian) | MR | Zbl

[7] Belogrivov I. I., “On transcedence and algebraic independence of the values of Kummer's function”, Sibirsk. Mat. Zh. (Siberian Mathematical Journal), 12:5 (1979), 961–982 (Russian)

[8] Ivankov P. L., “On the linear independence of the values of some functions”, Fundam. Prikl. Math. (Journal of Mathematical Sciences (New York)), 1:1 (1995), 191–206 (Russian) | MR

[9] Ivankov P. L., “On the arithmetic properties of the values of hypergeometric functions”, Matematicheskiy sbornik (Sbornik: Mathematics), 182:2 (1991), 283–302 (Russian)

[10] Galochkin A. I., “On arithmetic properties of the values of certain entire hypergeometric functions”, Sibirsk. Mat. Zhurnal (Siberian Mathematical Journal), XVII:6 (1976), 1220–1235 (Russian) | Zbl

[11] Galochkin A. I., “On some arithmetic properties of the coefficients of Kummer's function”, Fundam. Prikl. Mat. (Journal of Mathematical Sciences (New York)), 11:6 (2005), 27–32 (Russian) | Zbl

[12] Ivankov P. L., “On the values of hypergeometric functions with different irrational parameters”, Fundam. Prikl. Mat. (Journal of Mathematical Sciences (New York)), 11:6 (2005), 65–72 (Russian) | MR | Zbl

[13] Galochkin A. I., “Lower estimates of the linear forms in the values of some hypergeometric functions”, Mat. Zametki (Mathematical Notes), 8:1 (1970), 19–28 (Russian) | MR

[14] Galochkin A. I., “On diophantine approximations of the values of some entire functions with algebraic coefficients, I”, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (Moscow Univ. Math. Bulletin), 1978, no. 6, 25–32 (Russian) | MR

[15] Ivankov P. L., “On the calculation of the constants involved in the estimates of linear forms”, Izvestija vuzov. Mat. (Russian Mathematics (Izvestija VUZ. Mathematika)), 2000, no. 1(452), 31–36 (Russian)