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Transactions of the American Mathematical Society

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

 

 

On differential rings of entire functions


Authors: A. H. Cayford and E. G. Straus
Journal: Trans. Amer. Math. Soc. 209 (1975), 283-293
MSC: Primary 30A98
MathSciNet review: 0382671
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Abstract: Consider an entire function $ f$ which is a solution of the differential equation

$\displaystyle [{c_0}(z) + {c_1}(z)D + \ldots + {c_m}(z){D^m}]({f^n}) = P(f,f', \ldots ,{f^{(k)}})$

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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DOI: https://doi.org/10.1090/S0002-9947-1975-0382671-1
Keywords: Integer valued entire function, linear differential operator, approximation by algebraic integers, growth rate
Article copyright: © Copyright 1975 American Mathematical Society