Concordant sequences and integral-valued entire functions

Jonathan Pila; Fernando Rodriguez Villegas

Acta Arithmetica (1999)

  • Volume: 88, Issue: 3, page 239-268
  • ISSN: 0065-1036

Abstract

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A classic theorem of Pólya shows that the function 2 z is the “smallest” integral-valued entire transcendental function. A variant due to Gel’fond applies to entire functions taking integral values on a geometric progression of integers, and Bézivin has given a generalization of both results. We give a sharp formulation of Bézivin’s result together with a further generalization.

How to cite

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Jonathan Pila, and Fernando Rodriguez Villegas. "Concordant sequences and integral-valued entire functions." Acta Arithmetica 88.3 (1999): 239-268. <http://eudml.org/doc/207245>.

@article{JonathanPila1999,
abstract = {A classic theorem of Pólya shows that the function $2^z$ is the “smallest” integral-valued entire transcendental function. A variant due to Gel’fond applies to entire functions taking integral values on a geometric progression of integers, and Bézivin has given a generalization of both results. We give a sharp formulation of Bézivin’s result together with a further generalization.},
author = {Jonathan Pila, Fernando Rodriguez Villegas},
journal = {Acta Arithmetica},
keywords = {concordant sequences; entire functions; diffuse sequences},
language = {eng},
number = {3},
pages = {239-268},
title = {Concordant sequences and integral-valued entire functions},
url = {http://eudml.org/doc/207245},
volume = {88},
year = {1999},
}

TY - JOUR
AU - Jonathan Pila
AU - Fernando Rodriguez Villegas
TI - Concordant sequences and integral-valued entire functions
JO - Acta Arithmetica
PY - 1999
VL - 88
IS - 3
SP - 239
EP - 268
AB - A classic theorem of Pólya shows that the function $2^z$ is the “smallest” integral-valued entire transcendental function. A variant due to Gel’fond applies to entire functions taking integral values on a geometric progression of integers, and Bézivin has given a generalization of both results. We give a sharp formulation of Bézivin’s result together with a further generalization.
LA - eng
KW - concordant sequences; entire functions; diffuse sequences
UR - http://eudml.org/doc/207245
ER -

References

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  1. [1] J.-P. Bézivin, Itération de polynômes et fonctions entières arithmétiques, Acta Arith. 68 (1994), 11-25. MR 95k:30057. 
  2. [2] J.-P. Bézivin, Suites d'entiers et fonctions entières arithmétiques, Ann. Fac. Sci. Toulouse Math. 3 (1994), 313-334. MR 96a:11098. 
  3. [3] R. P. Boas, Comments on [15], in: George Pólya: Collected Papers, Vol. 1, R. P. Boas (ed.), M.I.T. Press, Cambridge, 1974, 771-773. 
  4. [4] N. Bourbaki, Commutative Algebra, Hermann, Paris, 1972. 
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  7. [7] K. Ford, personal communication of 3 April 1998. 
  8. [8] A. O. Gel’fond, Sur les fonctions entières, qui prennent des valeurs entières dans les points β n , β est un nombre entier positif et n = 1,2,3,..., Mat. Sb. 40 (1933), 42-47 (in Russian; French summary). Zbl0007.12102
  9. [9] A. O. Gel'fond, Calculus of Finite Differences, authorised English translation of the third Russian edition, Hindustan Publishing Corporation, Delhi, 1971. 
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  14. [14] C. Pisot, Über ganzwertige ganze Funktionen, Jahresber. Deutsch. Math.-Verein. 52 (1942), 95-102. Math. Rev. 4, p. 270. Zbl68.0162.02
  15. [15] G. Pólya, Ueber ganzwertige ganze Funktionen, Rend. Circ. Mat. Palermo 40 (1915), 1-16. Also in: Collected Papers, Vol. 1, R. P. Boas (ed.), M.I.T. Press, Cambridge, 1974, 1-16. Zbl45.0655.02
  16. [16] G. Pólya, Über ganze ganzwertige Funktionen, Nachr. Ges. Wiss. Göttingen 1920, 1-10. Also in: Collected Papers, Vol. 1, 131-140. Zbl47.0299.02
  17. [17] R. M. Robinson, Integer-valued entire functions, Trans. Amer. Math. Soc. 153 (1971), 451-468. Zbl0212.42201
  18. [18] A. Selberg, Über ganzwertige ganze transzendente Funktionen, Archiv for Math. og Naturvidenskab B. 44 (1941), 45-52. Also in: Collected Papers, Vol. 1, Springer, Berlin, 1989, 54-61. 
  19. [19] E. C. Titchmarsh, The Theory of Functions, 2nd ed., Oxford Univ. Press, 1939. Zbl0022.14602

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