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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
Citations:
Zbl 0405.10024; Zbl 0544.10034
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