Geodesic
On the Algebraic Independence of Values of Generalized Hypergeometric Functions
Matematičeskie zametki, Tome 94 (2013) no. 1, pp. 94-108.

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We consider hypergeometric functions satisfying homogeneous linear differential equations of arbitrary order. We prove general theorems on the algebraic independence of the solutions of a set of hypergeometric equations as well as of the values of these solutions at algebraic points. The conditions of most theorems are necessary and sufficient.
Keywords: generalized hypergeometric function, linear differential equation, algebraic independence of solutions, differential field, transcendence degree, contiguous functions.
Mots-clés : Galois group 
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V. A. Gorelov. On the Algebraic Independence of Values of Generalized Hypergeometric Functions. Matematičeskie zametki, Tome 94 (2013) no. 1, pp. 94-108. http://geodesic.mathdoc.fr/item/MZM_2013_94_1_a7/

[1] Yu. Lyuk, Spetsialnye matematicheskie funktsii i ikh approksimatsii, Mir, M., 1980 | MR | Zbl

[2] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii. Gipergeometricheskaya funktsiya. Funktsii Lezhandra, Spravochnaya matematicheskaya biblioteka, Nauka, M., 1965 | MR | Zbl

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

[4] A. B. Shidlovskii, Transtsendentnye chisla, Nauka, M., 1987 | MR | Zbl

[5] V. Kh. Salikhov, “Neprivodimost gipergeometricheskikh uravnenii i algebraicheskaya nezavisimost znachenii $\mathrm E$-funktsii”, Acta Arith., 53:5 (1990), 453–471 | MR | Zbl

[6] F. Beukers, W. D. Brownawell, G. Heckman, “Siegel normality”, Ann. of Math. (2), 127:2 (1988), 279–308 | MR | Zbl

[7] M. A. Cherepnev, “Ob algebraicheskoi nezavisimosti znachenii gipergeometricheskikh $E$-funktsii”, Matem. zametki, 57:6 (1995), 896–912 | MR | Zbl

[8] N. M. Katz, Exponential Sums and Differential Equations, Ann. of Math. Stud., 124, Princeton Univ. Press, Princeton, NJ, 1990 | MR | Zbl

[9] F. Beukers, “Some new results on algebraic independence of $\mathrm E$-functions”, New Advances in Transcendence Theory, Cambridge Univ. Press, Cambridge, 1988, 56–67 | MR | Zbl

[10] I. Kaplanskii, Vvedenie v differentsialnuyu algebru, IL, M., 1959 | MR | Zbl

[11] E. R. Kolchin, “Algebraic matric groups and the Picard-Vessiot theory of homogeneous linear ordinary differential equations”, Ann. of Math. (2), 49:1 (1948), 1–42 | MR | Zbl

[12] E. L. Ains, Obyknovennye differentsialnye uravneniya, Kharkov, 1939 | Zbl

[13] Dzh. Sansone, Obyknovennye differentsialnye uravneniya, T. 1, IL, M., 1953 | MR | Zbl

[14] V. A. Gorelov, “Ob algebraicheskikh tozhdestvakh mezhdu obobschennymi gipergeometricheskimi funktsiyami”, Matem. zametki, 88:4 (2010), 511–516 | DOI | MR | Zbl

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

[16] G. G. Viskina, V. Kh. Salikhov, “Algebraicheskie sootnosheniya mezhdu gipergeometricheskoi $E$-funktsiei i ee proizvodnymi”, Matem. zametki, 71:6 (2002), 832–844 | DOI | MR | Zbl

[17] V. Kh. Salikhov, “Formalnye resheniya lineinykh differentsialnykh uravnenii i ikh primenenie v teorii transtsendentnykh chisel”, Tr. MMO, 51, Izdatelstvo Mosk. un-ta, M., 1988, 223–256 | MR | Zbl

[18] E. Kamke, Spravochnik po obyknovennym differentsialnym uravneniyam, Nauka, M., 1961 | MR | Zbl