## Lower bounds for polynomials in the values of certain entire functions.(English. Russian original)Zbl 0878.11030

Sb. Math. 187, No. 12, 1791-1818 (1996); translation from Mat. Sb. 187, No. 12, 57-86 (1996).
Let $$f_1(z),\dots, f_m(z)$$ $$(m\geq 2)$$ be a set of $$E$$-functions satisfying a system of first order homogeneous linear differential equations. Suppose that the functions in a certain matrix of fundamental solutions of this system of differential equations are homogeneously algebraically independent over $$\mathbb{C}(z)$$, that $$\alpha\in\mathbb{Q}\setminus \{0\}$$ is not a singular point, and that $$d\in\mathbb{N}$$. The author proves that there exist positive constants $$\gamma= \gamma(f_1,\dots,f_m; \alpha,d)$$ and $$C=C(f_1,\dots, f_m;\alpha,d)$$ such that for any homogeneous polynomial $$P\in \mathbb{Z}[y_1,\dots, y_m]$$ of degree $$d$$, the inequality $|P(f_1(\alpha),\dots, f_m(\alpha))|> C|h_1\dots h_w|^{-1} H^{1-\gamma(\log\log H)^{1/(m^2- m+2)}},$
$H=\max_{1\leq i\leq w}|h_i|\geq 3,$ holds, where $$h_1,\dots, h_w$$ are all nonzero coefficients of $$P$$. From this result we can immediately get lower estimates for linear forms of $$f_1(\alpha),\dots, f_m(\alpha)$$. The author also gives a more general theorem on lower estimates for polynomials of the values of $$E$$-functions, which implies the above result, introducing a new ideal based on Padé approximation. Furthermore, as an application of the result, the author considers Siegel’s classical example $$K_\lambda(z)$$.

### MSC:

 11J82 Measures of irrationality and of transcendence 41A21 Padé approximation 11J91 Transcendence theory of other special functions 33C60 Hypergeometric integrals and functions defined by them ($$E$$, $$G$$, $$H$$ and $$I$$ functions)
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