On a refinement of the Schneider–Lang theorem. II. The arithmetic of the degenerate case

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 $g_3$).