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 which is a solution of the differential equation

THEOREM. *If satisfies the equation above then is of finite type in case and of finite exponential order in case *.

We use this result to prove a conjecture made in [2] that entire functions of order , all of whose derivatives at points are integers in an imaginary quadratic number field, must be solutions of linear differential equations with constant coefficients and therefore of order .

**[1]**Enrico Bombieri,*Algebraic values of meromorphic maps*, Invent. Math.**10**(1970), 267–287. MR**0306201****[2]**Afton H. Cayford,*A class of integer valued entire functions*, Trans. Amer. Math. Soc.**141**(1969), 415–432. MR**0244486**, 10.1090/S0002-9947-1969-0244486-1**[3]**B. Ja. Levin,*Distribution of zeros of entire functions*, American Mathematical Society, Providence, R.I., 1964. MR**0156975****[4]**E. G. Straus,*Differential rings of meromorphic functions*, Acta Arith.**21**(1972), 271–284. MR**0308418****[5]**-,*Differential rings of analytic functions of a nonarchimedean variable*, Diophantine Approximation and its Applications, Academic Press, New York, 1973, pp. 295-308.

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Additional Information

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