On algebraic identities between solution matrices of generalized hypergeometric equations

V. A. Gorelov

Čebyševskij sbornik, Tome 21 (2020) no. 1, pp. 135-144.

Résumé

The examples of algebraic identities between solution matrices of generalized hypergeometric equations are found in paper. These identities generate all the algebraic identities between components of solutions of hypergeometric equations in some cases. Generalized hypergeometric functions (see [1–5]) are defined as $$ {}_l\varphi_{q}(z)={}_l\varphi_{q}(\vec \nu;\vec\lambda;z)= {}_{l+1}F_{q}\left(\left.{1,\nu_1,\dots,\nu_l\atop\lambda_1,\dots,\lambda_q}\right|z\right)= \sum_{n=0}^\infty \frac{(\nu_1)_n\dots (\nu_l)_n}{(\lambda_1)_n \dots(\lambda_{q})_n} z^n, $$ where $0\leqslant l\leqslant q$, $(\nu)_0=1$, $(\nu)_n=\nu(\nu+1) \dots (\nu+n-1)$, $\vec\nu=(\nu_1,\dots,\nu_l)\in {\mathbb C}^l$, $\vec \lambda\in ({\mathbb C}\setminus{\mathbb Z^-})^q$. The function ${}_l\varphi_{q}(\vec \nu;\vec\lambda;z)$ satisfies the (generalized) hypergeometric differential equation $$ {L}(\vec \nu;\vec\lambda;z) y =(\lambda_1-1)\dots(\lambda_q-1), $$ where $$ {L}(\vec \nu;\vec\lambda;z) \equiv \prod_{j=1}^q(\delta+\lambda_j-1)- z\prod_{k=1}^l(\delta+\nu_k), \delta=z\frac{d}{dz}. $$ The Siegel-Shidlovskii method (see [4], [5]) is one of the main methods in the theory of transcendental numbers. It permits to establish the transcendency and the algebraic independence of the values of entire functions of some class, which contains the functions ${}_l\varphi_{q}(\alpha z^{q-l})$, provided that these functions are algebraically independent over ${\mathbb C}(z)$. F. Beukers, W.D. Brownawell and G. Heckman introduced in paper [6] notions of cogredience and contragredience of differential equations, which are important for determination of algebraic dependence and independence of functions (these notions appeared firstly in paper [7] of E. Kolchin really). This work contains detailed proof and further development of results connected with cogredience and contragredience, that have been published in notes [8], [9]. Some results in [6] have been revised particularly.

Détail

Keywords: hypergeometric functions, Siegel's method, algebraic independence.

Bibliographie

[1] Luke Y., Mathematical functions and their approximations, Academic Press, New York, 1975 | MR | Zbl

[2] Kratzer A., Franz W., Transzendente Funktionen, Akademische Verlagsgesellschaft, Leipzig, 1960 (German) | MR | Zbl

[3] Andrews G., Askey R., Roy R., Special Functions, Cambridge University Press, 2000 | MR | Zbl

[4] Siegel C. L., “Über einige Anwendungen Diophantischer Approximationen”, Abh. Preuss. Acad. Wiss., Phys. Math. Kl., no. 1, 1929–1930 (German)

[5] Shidlovskii A.B., Transcendental numbers, Walter de Gruyter, Berlin–New York, 1989 | MR | MR | Zbl

[6] Beukers F., Brownawell W. D., Heckman G., “Siegel normality”, Annals of Math., 127 (1988), 279–308 | DOI | MR | Zbl

[7] Kolchin E. R., “Algebraic groups and algebraic dependence”, Amer. J. Math., 90:4 (1968), 1151–1164 | DOI | MR | Zbl

[8] Gorelov V.A., “On algebraic identities between generalized hypergeometric functions”, Mathematical Notes, 88:4 (2010), 487–491 | DOI | MR | Zbl

[9] Gorelov V.A., Siberian Electronic Mathematical Reports, 16 (2019), 258–262 (Russian) | DOI | MR | Zbl

[10] Gorelov V.A., “About new algebraic identities between generalized hypergeometric functions”, Vestnik Moskovskogo energeticheskogo instituta, 2008, no. 6, 129–138 (Russian)

[11] Gorelov V.A., “On the algebraic independence of values of generalized hypergeometric functions”, Mathematical Notes, 94:1 (2013), 82–95 | DOI | MR | Zbl

[12] Beukers F., Jouhet F., “Duality relations for hypergeometric series”, Bulletin of the London Math. Soc., 47 (2015), 343–358 | DOI | MR | Zbl

[13] Feng R., Kuznetsov A., Yang F., “A short proof of duality relations for hypergeometric functions”, J. of Math. Analysis and Applications, 443 (2016), 116–122 | DOI | MR | Zbl

[14] Gorelov V. A., “On contiguity relations for generalized hypergeometric functions”, Problemy Analiza – Issues of Analysis, 7(25):2 (2018), 39–46 | DOI | MR | Zbl

[15] Sansone G., Equazioni differenziali nel campo reale, Bologna, 1948 (Italian)

[16] Cherepnev M.A., “Algebraic independence of values of hypergeometric E-functions”, Mathematical Notes, 57:6 (1995), 630–642 | DOI | MR | Zbl