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Fonctions Generatrices Transcendantes a Coefficients Engendres par Automates

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STACS 88 (STACS 1988)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 294))

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Résumé

Soit (u(n)) une suite de ±1 engendrée par un 2-automate, nous montrons que la fonction génératrice associée

$$f (Z) = \sum\limits_{n = 0}^\infty {u(n)} Z^n$$

a des propriétés généralisant celles de la fonction de Thue-Morse:

Il existe un vecteur F(Z) dont une des composantes est f(Z) et une matrice carrée M(Z) tels que l'on ait: F(Z)=M(Z)F(Z2) d'où une expression de F par un produit infini (matriciel). De plus nous donnons des développements en séries de fractions rationnelles pour f. Enfin, dans le cas particulier de la fonction de Thue-Morse, nous montrons que tous les points du cercle unité sont des singularités de type non algébrique.

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References

  1. J.P. ALLOUCHE, Théorie des nombres et automates, Thèse, Bordeaux, 1983.

    Google Scholar 

  2. J.P. ALLOUCHE, Automates finis en théorie des nombres, à paraître dans Expo. Math., 1987.

    Google Scholar 

  3. J.M. AUTEBERT, P. FLAJOLET, J. GABARRO, Prefixes of infinite words and ambiguous context-free languages, Inform. Process. Lett. 25, 4, 1987, p. 211–216.

    Google Scholar 

  4. M. BALAZARD, Communication privée.

    Google Scholar 

  5. J. BERSTEL, Every iterated morphism yields a Co-CFL, Inform. Process. Lett. 22, 1, 1986, p. 7–9.

    Google Scholar 

  6. J. BEAUQUIER, L. THIMONIER, Prefix-free words of length n over m letters: two-sided well balanced parentheses and palindroms, Colloque de combinatoire énumérative U.Q.A.M. 1985, Canada, Lect. Notes in Math., 1234, p. 27–33.

    Google Scholar 

  7. G. CHRISTOL, T. KAMAE, M. MENDES FRANCE et G. RAUZY, Suites algébriques, automates et substitutions, Bull. Soc. Math. France, 108, 1980, p. 401–419.

    Google Scholar 

  8. L. COMTET, Analyse combinatoire, P.U.F., Paris 1970.

    Google Scholar 

  9. F.M. DEKKING, Transcendance du nombre de Thue-Morse, C.R. Acad. Sci., sér. A, t. 285, 1977, p. 157–160.

    Google Scholar 

  10. P. DUMAS, L. THIMONIER, On generating functions of the language of the prefix-free palindroms over m letters, Rapport L.R.I., 1987.

    Google Scholar 

  11. S. EILENBERG, Automata, languages and machines, vol. A, Academic Press, 1976.

    Google Scholar 

  12. P. FLAJOLET, Analytic models and ambiguity of context-free languages, Theoretical Computer Science 49, 1987, p. 283–309.

    Google Scholar 

  13. P. FLAJOLET and N. MARTIN, Probabilistic counting algorithms and data base applications, J. Computer System Sc., 31, 1985, p. 182–209.

    Google Scholar 

  14. P. FLAJOLET, M. REGNIER, D. SOTTEAU, Algebraic methods for trie statistics, Ann. Discr. Math., 25, 1985, p. 145–188.

    Google Scholar 

  15. P. FLAJOLET, R. SEDGEWICK, Mathematical analysis of algorithms, Cours de l'Université de Princeton, U.S.A., 1986.

    Google Scholar 

  16. J.H. LOXTON and A. J. Van Der POORTEN, Arithmetic properties of the solutions of a class of functional equations, J. Reine Angew. Math., 330, 1982, p. 159–172.

    Google Scholar 

  17. G. POLYA, Collected papers, Vol. 1, Singularities of analytic functions, R.P. Boas ed., the M.I.T. Press, Cambridge, 1974.

    Google Scholar 

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Author information

Authors and Affiliations

  1. U.A. 226, Math. et Info., 351 Cours de la Libération, 33405, Talence Cedex

    Jean-Paul Allouche

  2. Lycée Louis Le Grand, 123 Rue Saint-Jacques, 75005, Paris

    Bernard Rande

  3. L.R.I., U.A. 410, Université de Paris-Sud Orsay, Bât. 490, France

    Loÿs Thimonier

Authors
  1. Jean-Paul Allouche
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  2. Bernard Rande
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  3. Loÿs Thimonier
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Editor information

Robert Cori Martin Wirsing

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© 1988 Springer-Verlag Berlin Heidelberg

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Allouche, JP., Rande, B., Thimonier, L. (1988). Fonctions Generatrices Transcendantes a Coefficients Engendres par Automates. In: Cori, R., Wirsing, M. (eds) STACS 88. STACS 1988. Lecture Notes in Computer Science, vol 294. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0035843

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  • DOI: https://doi.org/10.1007/BFb0035843

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-18834-6

  • Online ISBN: 978-3-540-48190-4

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