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ArticleOriginal scientific text

Title

Concordant sequences and integral-valued entire functions

Authors 1, 2

Affiliations

  1. Department of Mathematics, University of Melbourne, Melbourne, Australia
  2. Department of Mathematics, University of Texas at Austin, Austin, Texas 78712-1082, U.S.A.

Abstract

A classic theorem of Pólya shows that the function 2z 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.

Bibliography

  1. J.-P. Bézivin, Itération de polynômes et fonctions entières arithmétiques, Acta Arith. 68 (1994), 11-25. MR 95k:30057.
  2. J.-P. Bézivin, Suites d'entiers et fonctions entières arithmétiques, Ann. Fac. Sci. Toulouse Math. 3 (1994), 313-334. MR 96a:11098.
  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. N. Bourbaki, Commutative Algebra, Hermann, Paris, 1972.
  5. R. C. Buck, Integral valued entire functions, Duke Math. J. 15 (1948), 879-891.
  6. P. Bundschuh, A theorem of Gelfond via Schneider's method, in: New Trends in Probability and Statistics, F. Schweiger and E. Mantavicius (eds.), VSP, Utrecht, 1992, 9-15.
  7. K. Ford, personal communication of 3 April 1998.
  8. A. O. Gel'fond, Sur les fonctions entières, qui prennent des valeurs entières dans les points βn, β est un nombre entier positif et n = 1,2,3,..., Mat. Sb. 40 (1933), 42-47 (in Russian; French summary).
  9. A. O. Gel'fond, Calculus of Finite Differences, authorised English translation of the third Russian edition, Hindustan Publishing Corporation, Delhi, 1971.
  10. R. R. Hall, On pseudopolynomials, Mathematika 8 (1971), 71-77.
  11. G. H. Hardy, On a theorem of Mr. G. Pólya, Proc. Cambridge Philos. Soc. 19 (1917), 60-63.
  12. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Clarendon Press, Oxford, 1979.
  13. A. Perelli and U. Zannier, On recurrent mod p sequences, J. Reine Angew. Math. 348 (1984), 135-146. Math. Rev. 85g:11012.
  14. C. Pisot, Über ganzwertige ganze Funktionen, Jahresber. Deutsch. Math.-Verein. 52 (1942), 95-102. Math. Rev. 4, p. 270.
  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.
  16. G. Pólya, Über ganze ganzwertige Funktionen, Nachr. Ges. Wiss. Göttingen 1920, 1-10. Also in: Collected Papers, Vol. 1, 131-140.
  17. R. M. Robinson, Integer-valued entire functions, Trans. Amer. Math. Soc. 153 (1971), 451-468.
  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. E. C. Titchmarsh, The Theory of Functions, 2nd ed., Oxford Univ. Press, 1939.
Acta Arithmetica
1999/88/3
Export to PBN, Opens in new tab
Pages:
239-268
Main language of publication
English
Received
1998-05-14
Accepted
1998-10-16
Published
1999
Exact and natural sciences