# Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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## On differential rings of entire functionsHTML articles powered by AMS MathViewer

by A. H. Cayford and E. G. Straus
Trans. Amer. Math. Soc. 209 (1975), 283-293 Request permission

## 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 $\leqslant 1$.
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