On the values of hypergeometric function with parameter from quadratic field

P. L. Ivankov

Čebyševskij sbornik, Tome 20 (2019) no. 2, pp. 178-185.

Résumé

In order to investigate arithmetic properties of the values of generalized hypergeometric functions with rational parameters one usually applies Siegel's method. By means of this method have been achieved the most general results concerning the above mentioned properties. The main deficiency of Siegel's method consists in the impossibility of its application for the hypergeometric functions with irrational parameters. In this situation the investigation is usually based on the effective construction of the functional approximating form (in Siegel's method the existence of that form is proved by means of pigeon-hole principle). The construction and investigation of such a form is the first step in the complicated reasoning which leads to the achievement of arithmetic result.Applying effective method we encounter at least two problems which make extremely narrow the field of its employment. First, the more or less general effective construction of the approximating form for the products of hypergeometric functions is unknown. While using Siegel's method one doesn't deal with such a problem. Hence the investigator is compelled to consider only questions of linear independence of the values of hypergeometric functions over some algebraic field. Choosing this field is the second problem. The great majority of published results concerning corresponding questions deals with imaginary quadratic field (or the field of rational numbers). Only in exceptional situations it is possible to investigate the case of some other algebraic field.We consider here the case of a real quadratic field. By means of a special technique we establish linear independence of the values of some hypergeometric function with irrational parameter over such a field.

Détail

Keywords: hypergeometric function, effective construction, linear independence, real quadratic field.

Bibliographie

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

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

[3] Shidlovskii A. B., Transcendental numbers, Nauka, M., 1987, 448 pp. (Russian) | MR

[4] Shidlovskii A. B., “On transcendentality and algebraic independence of the values of entire functions of certain class”, Dokl. Akad. Nauk SSSR, 96:4 (1954), 697–700 (Russian) | Zbl

[5] Shidlovskii A. B., “Transcendence and algebraic independence of values of $E$-functions satisfying linear nonhomogeneous differential equations of the second order”, Dokl. Akad. Nauk SSSR, 169:1 (1954), 42–45 (Russian)

[6] Shidlovskii A. B., “Algebraic independence of the values of certain hypergeometric $E$-functions”, Trudy Moskov. Mat. Obsh., 18, no. 4, 1968, 55–64 (Russian) | MR

[7] Belogrivov I. I., “On transcendence and algebraic independence of values of certain hypergeometric $E$-functions”, Dokl. Akad. Nauk SSSR, 174:2 (1967), 267–270 (Russian) | MR | Zbl

[8] Chirsky V. G., “On arithmetic properties of the values of hypergeometric functions with irrational parameters”, Vestnik Moskov. Univ. Ser. 1. Mat. Meh., 1978, no. 5, 3–8 | Zbl

[9] Salikhov V. Kh., “Irreducibility of hypergeometric equations and algebraic independence of values of $E$-functions”, Acta Arithm., 53:5 (1990), 453–471 | MR | Zbl

[10] Cherepnev M. A., “On algebraic independence of values of hypergeometric $E$-functions”, Mat. Zametki, 57:6 (1995), 896–912 | MR | Zbl

[11] Salikhov V. Kh., “Criterion for the algebraic independence of the values of hypergeometric $E$-functions (even case)”, Mat. Zametki, 64:2 (1998), 273–284 | Zbl

[12] Gorelov V. A., “On algebraic independence of the values of hypergeometric functions”, Mat. Zametki, 94:1 (2013), 94–108 | Zbl

[13] Gorelov V. A., “On algebraic properties of the solutions of nonhomogeneous hypergeometric equations”, Mat. Zametki, 99:5 (2016), 658–672 | MR | Zbl

[14] Ch. F. Osgood, “Some theorems on diophantine approximation”, Trans. Amer. Math. Soc., 123:1 (1966), 64–87 | MR | Zbl

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

[16] Galochkin A. I., “Sharpening of the estimates of some linear forms”, Mat. Zametki, 20:1 (1976), 35–45 (Russian) | MR | Zbl

[17] Galochkin A. I., “On arithmetic properties of the values of some entire hypergeometric functions”, Sibirsk. Mat. Zh., 17:6 (1976), 1220–1235 (Russian) | MR | Zbl

[18] Galochkin A. I., “Estimates, unimprovable with respect to height, for certain linear forms”, Mat. Sb, 124(166):3 (1984), 416–430 (Russian) | MR | Zbl

[19] Korobov A. N., “Estimates of some linear forms”, Vestnik Moskov. Univ. Ser. I Mat. Meh., 1983, no. 6, 36–41 (Russian)

[20] Popov A. Yu., “Approximations of some degrees of the number $e$”, Diophantovy priblizhenija, v. 1, Moskov. Gos. Univ., M., 1985 (Russian)

[21] Ivankov P. L., “On approximation of the values of some functions”, Vestnik Moskov. Univ. Ser. I Mat. Meh., 1994, no. 4, 12–15 (Russian) | MR | Zbl

[22] Ivankov P. L., “On simultaneous approximations of the values of some entire functions by the numbers from a cubic field”, Vestnik Moskov. Univ. Ser. 1, Mat. Meh., 1987, no. 3, 53–56 (Russian) | Zbl

[23] Ivankov P. L., “On linear independence of values of entire hypergeometric functions with irrational parameters”, Sibirsk. Mat. Zh., 34:5 (1993), 839–847 | MR | Zbl

[24] Ivankov, P. L., “On approximation of the values of hypergeometric function with a parameter from real quadratic field”, Mathematics and Mathematical Modelling, 2017, no. 1, 25–33 (Russian)

[25] Markushevich A. I., Theory of analytic functions, v. I, “Nauka”, M., 1967, 486 pp. (Russian) | MR