On differential rings of entire functions

Abstract:

Consider an entire function $f$ which is a solution of the differential equation \[ [{c_0}(z) + {c_1}(z)D + \ldots + {c_m}(z){D^m}]({f^n}) = P(f,f’, \ldots ,{f^{(k)}})\] where ${c_i}(z)$ are entire functions in a differential ring $R$ and $P$ is a polynomial in a differential field related to $R$. We prove the following THEOREM. If $f$ satisfies the equation above then $f$ is of finite type in case $R = {\mathbf {C}}$ and of finite exponential order in case $R = {\mathbf {C}}[z]$. We use this result to prove a conjecture made in [2] that entire functions of order $\rho < s$, all of whose derivatives at $s$ points are integers in an imaginary quadratic number field, must be solutions of linear differential equations with constant coefficients and therefore of order $\leqslant 1$.