On the Cogredience and Contragredience of Linear Differential Equations and Systems
DOI:
https://doi.org/10.24160/1993-6982-2021-6-148-151Keywords:
Siegel’s method, algebraic independence, E-functionsAbstract
The Siegel-Shidlovskii method remains one of the basic methods in the theory of transcendental numbers. By using this method, it is possible to prove the transcendence and algebraic independence of the values of entire functions of a certain class (so-called E-functions). A necessary condition for applying this method is that all of the considered functions must constitute a solution of a system of linear differential equations and were algebraically independent over .
The question about algebraic independence of the solutions of linear differential equations and systems of such equations is of great importance in differential algebra, analytical theory of differential equations, theory of special functions, and calculus (in the broad sense of the word). As is shown in papers by E. Kolchin, F. Beukers, W.D. Brownawell, and G. Heckman, this question boils down in many instances to verification of the cogredience and contragredience condition.
Two systems of 1st order linear homogeneous differential equations with coefficients from are said to be cogredient (or, respectively, contragredient), if for arbitrary fundamental matrices and Ψ of these systems one of the equations Φ=gBΨC, Φ (ΨC)^*=gB,
is fulfilled, where C∈GL(C), B∈GL(C(z)), g=g(z) is a function with the condition g^'/g∈C(z), and A^* is the matrix transposed to . The notions of cogredience and contragredience for linear homogeneous differential equations of arbitrary order are defined similarly.
Another, more restricted definitions of cogredience and contragredience were in fact used in some papers of the author, devoted to generalized hypergeometric functions. According to these definitions, the function in the presented equalities is the product of a power function and an exponential function of some kind. The conditions for equivalence of these definitions are found.
References
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Для цитирования: Горелов В.А. О коградиентности и контрградиентности линейных дифференциальных уравнений и систем // Вестник МЭИ. 2021. № 6. С. 148—151. DOI: 10.24160/1993-6982-2021-6-148-151
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Работа выполнена при поддержке: Министерства науки и высшего образования РФ (проект № FSWF-2020-0022)
The work is executed at support: Ministry of Science and Higher Education of the Russian Federation (Project No. FSWF-2020-0022)
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1. Beukers F., Brownawell W.D., Heckman G. Siegel Normality. Annals Math. 1988;127:279—308.
2. Kolchin E.R. Algebraic Groups and Algebraic Dependence. Amer. J. Math. 1968;90;4:1151—1164.
3. Beukers F. Some New Results on Algebraic Independence of E-functions. New Advances in Transcendence Theory. Cambridge: Cambridge Univ. Press., 1988:56—67.
4. Katz N.M. Exponential Sums and Differential Equations. Princeton: Princeton Univ. Press, 1990.
5. Siegel C.L. Uber Einige Anwendungen Diophantischer Approximationen. Abh. Preuss. Acad. Wiss. Phys.-math. Kl. 1929—1930;1:1—70.
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11. Gorelov V.A. On Contiguity Relations for Generalized Hypergeometric Functions. Problemy Analiza — Issues of Analysis. 2018;7(25);2:39—46.
12. Nesterenko Yu.V. Priblizheniya Ermita – Pade Obobshchennykh Gipergeometricheskikh Funktsiy. Matem. sbornik. 1994;185;10:39—72. (in Russian).
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14. Gorelov V.A. Otsenki Mnogochlenov ot Znacheniy E-funktsiy. Vestnik MEI. 2020;4:136—143. (in Russian)
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For citation: Gorelov V.A. On the Cogredience and Contragredience of Linear Differential Equations and Systems. Bulletin of MPEI. 2021;6:148—151. (in Russian). DOI: 10.24160/1993-6982-2021-6-148-151
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The work is executed at support: Ministry of Science and Higher Education of the Russian Federation (Project No. FSWF-2020-0022)
