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On the Siegel conjecture for second-order homogeneous linear differential equations. (English. Russian original) Zbl 1062.34096
Math. Notes 75, No. 4, 513-529 (2004); translation from Mat. Zametki 75, No. 4, 549-565 (2004).
The paper deals with an investigation of a class of entire analytic functions (so-called \(E\)-functions), satisfying systems of linear differential equations from \(\mathbb{C}(z)\). Siegel conjectured that any \(E\)-function can be represented as a polynomial with suitable algebraic coefficients. In the present paper, it is proved that an \(E\)-function satisfies a second-order homogeneous linear differential equation with coefficients from \(\mathbb{C}(z)\) if and only if it is of a special form, which is described explicitly.

MSC:
34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
30D10 Representations of entire functions of one complex variable by series and integrals
34M25 Formal solutions and transform techniques for ordinary differential equations in the complex domain
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