Entire functions which are infinitely integer-valued at a finite number of points

Authors:
P. Lockhart and E. G. Straus

Journal:
Trans. Amer. Math. Soc. **293** (1986), 643-654

MSC:
Primary 30D15

DOI:
https://doi.org/10.1090/S0002-9947-1986-0816316-8

MathSciNet review:
816316

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper determines arithmetic limits for the growth rates of entire functions which are infinitely integer valued on a finite set . The characterization of such functions with growth rate less than the arithmetic limit is complete if there exist exponential polynomials which are infinitely integer valued on .

**[1]**Alan Baker,*Transcendental number theory*, Cambridge University Press, London-New York, 1975. MR**0422171****[2]**Afton H. Cayford,*A class of integer valued entire functions*, Trans. Amer. Math. Soc.**141**(1969), 415–432. MR**0244486**, https://doi.org/10.1090/S0002-9947-1969-0244486-1**[3]**A. H. Cayford and E. G. Straus,*On differential rings of entire functions*, Trans. Amer. Math. Soc.**209**(1975), 283–293. MR**0382671**, https://doi.org/10.1090/S0002-9947-1975-0382671-1**[4]**B. Ja. Levin,*Distribution of zeros of entire functions*, American Mathematical Society, Providence, R.I., 1964. MR**0156975****[5]**L. M. Milne-Thomson,*The Calculus of Finite Differences*, Macmillan and Co., Ltd., London, 1951. MR**0043339****[6]**L. D. Neidleman and E. G. Straus,*Functions whose derivatives at one point form a finite set*, Trans. Amer. Math. Soc.**140**(1969), 411–422. MR**0241644**, https://doi.org/10.1090/S0002-9947-1969-0241644-7**[7]**Daihachiro Sato,*Two counterexamples and some remarks on integer-valued functions*, Sûgaku**14**(1962/1963), 95–98 (Japanese). MR**0150302****[8]**Daihachiro Sato and Ernst G. Straus,*On the rate of growth of Hurwitz functions of a complex or 𝑝-adic variable*, J. Math. Soc. Japan**17**(1965), 17–29. MR**0192030**, https://doi.org/10.2969/jmsj/01710017**[9]**E. G. Straus,*On entire functions with algebraic derivatives at certain algebraic points*, Ann. of Math. (2)**52**(1950), 188–198. MR**0035822**, https://doi.org/10.2307/1969518**[10]**Ernst G. Straus,*On the polynomials whose derivatives have integral values at the integers*, Proc. Amer. Math. Soc.**2**(1951), 23–27. MR**0040481**, https://doi.org/10.1090/S0002-9939-1951-0040481-8**[11]**E. G. Straus,*Differential rings of meromorphic functions*, Acta Arith.**21**(1972), 271–284. MR**0308418**, https://doi.org/10.4064/aa-21-1-271-284**[12]**-,*Differential rings of meromorphic functions of a non-Archimedean variable, Diophantine approximation and its applications*(Proc. Conf. Washington, D. C., 1972), Academic Press, New York, 1973, pp. 295-308.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
30D15

Retrieve articles in all journals with MSC: 30D15

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1986-0816316-8

Article copyright:
© Copyright 1986
American Mathematical Society