Abstract
We prove a generalization of Shidlovskii’s theorem on the algebraic independence of the values ofE-functions satisfying a system of linear differential equations that is well known in the theory of transcendental numbers. We consider the case in which the values ofE-functions are taken at singular points of these systems. Using the obtained results, we prove Siegel’s conjecture that, for the case of first-order differential equations, anyE-function satisfying a linear differential equation is representable as a polynomial in hypergeometricE-functions.
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References
C. L. Siegel, “Über einige Anwendungen Diophantischer Approximationen,”Abh. Preuss. Akad. Wiss., Phys.-Math. Kl., No. 1, 1–70 (1929–1930).
A. B. Shidlovskii,Transcendental Numbers [in Russian], Nauka, Moscow (1987)
C. L. Siegel,Transcendental Numbers, Princeton Univ. Press, Princeton (1949).
A. I. Galochkin, “A criterion for hypergeometric Siegel’s functions to belong to the class ofE-functions,”Mat. Zametki [Math. Notes],29, No. 1, 3–14 (1981).
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Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 174–190, February, 2000.
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Gorelov, V.A. Algebraic independence of the values ofE-functions at singular points and Siegel’s conjecture. Math Notes 67, 138–151 (2000). https://doi.org/10.1007/BF02686240
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DOI: https://doi.org/10.1007/BF02686240