Abstract
Arithmetic properties of the values of meromorphic functions \(g_1(z),\dots,g_n(z)\) of finite order such that each derivative \(g'_i(z)\) depends algebraically on the functions \(g_1(z),\dots,g_n(z)\) over an algebraic number field \(K\) with \([K:\mathbb{Q}]<+\infty\) are considered. It is shown that if the transcendence degree of the field \(\mathbb{C}(g_1(z),\dots,g_n(z))\) equals 1 and there exists a \(z_0\in\mathbb{C}\) at which \(g_i(z_0)\in K\) for all \(i\) , then the functions \(g_i(z)\) are of one of the forms \(\{R_i(z-z_0)\}\) , \(\{R_i(e^{\alpha(z-z_0)})\}\) , and \(\bigl\{R_{i,1}\bigl(\wp(z-z_0+{\omega_1}/{2})\bigl)+ \wp'\bigl(z-z_0+{\omega_1}/{2}\bigl) R_{i,2}\bigl(\wp(z-z_0+{\omega_1}/{2})\bigl)\bigr\}\) (where all \(R_{i,j}(t)\) and \(R_i(t)\) are rational functions with coefficients in a field \(K_1\) such that \([K_1:K]<+\infty\) , \(\alpha\in K_1\) , and \(\wp(z)\) is the Weierstrass elliptic function one of whose period is \(\omega_1\) with algebraic (belonging to \(K_1\) ) invariants \(g_2\) and \(g_3\) ).
