On a~measure of irrationality for values of $G$-functions

W. V. Zudilin

Izvestiya. Mathematics , Tome 60 (1996) no. 1, pp. 91-118

Résumé

It is shown that values of $G$-functions satisfying a system of linear differential equations are irrational at rational points $a/b$ with $a\in\mathbb Z$ and $b\in\mathbb N$ such that $b>C(\varepsilon)|a|^{2+\varepsilon}$ for an arbitrary positive $\varepsilon$. In the case of a generalized polylogarithmic function $$ f(z)=\sum_{\nu=1}^\infty\frac{z^\nu}{(\nu+\lambda)^m}, \quad m\geqslant 2, \enskip \lambda\in\mathbb Q\setminus\{-1,-2,\dots\}, $$ an explicit form of $C(\varepsilon)$ is found.