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On rational approximations of values of a certain class of entire functions. (English. Russian original) Zbl 0848.11031
Sb. Math. 186, No. 4, 555-590 (1995); translation from Mat. Sb. 186, No. 4, 89-124 (1995).
The author proves the following main theorem: Suppose that \(\mathbb{Q} E\)-functions \(f_1(z),\dots, f_m(z)\) \((m\geq 2)\) are algebraically independent over \(\mathbb{C}(z)\) and satisfy the system of linear differential equations \(y_i'= Q_{i0}+ \sum^m_{\ell= 1} Q_{i\ell}(z) y_\ell\), \(i= 1,\dots, m\) with the coefficients in \(\mathbb{C}(z)\). Let \(\alpha\) be a nonzero rational number. Then there exists a positive number \(\gamma= \gamma(f_1,\dots, f_m, \alpha)\) such that for any integer \(q\) with \(|q|> q^*(f_1,\dots, f_m, \alpha)\), the inequalities \[ |f_\ell(\alpha)- p/q|> |q|^{- 2- \gamma(\ln \ln |q|)^{-1/(m+ 1)}},\quad \ell= 1,\dots, m \] hold. This main theorem gives some improvements of earlier work by A. B. Shidlovskii [J. Austral. Math. Soc., Ser. A 27, 385-407 (1979; Zbl 0405.10024)] and G. V. Chudnovsky [Proc. Natl. Acad. Sci. USA 81, 1926-1930 (1984; Zbl 0544.10034)].

MSC:
11J82 Measures of irrationality and of transcendence
11J91 Transcendence theory of other special functions
11J85 Algebraic independence; Gel’fond’s method
34M99 Ordinary differential equations in the complex domain
30D15 Special classes of entire functions of one complex variable and growth estimates
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