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

Zudilin, V. V.

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 \(K_\lambda(z)\).

Reviewer: Zhu Yaochen (Beijing)

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)

Keywords:

entire functions; generalized hypergeometric \(E\)-functions; lower estimates for linear forms; lower estimates for polynomials; Padé approximation