The Ramanujan Journal

, Volume 2, Issue 3, pp 387–410 | Cite as

Polynômes d'Euler et Fractions Continues de Stieltjes-Rogers

  • Dominique Dumont
  • Jiang Zeng
Article

Abstract

It is well-known that the Euler polynomials E2n(x) with n ≥ 0 can be expressed as a polynomial Hn(x(x − 1)) of x(x − 1). We extend Hn(u) to formal power series for n < 0 and prove several properties of the coefficients appearing in these polynomials or series, which generalize some recent results, independently obtained by Hammersley [7] and Horadam [8], and answer a question of Kreweras [9]. We also deduce several continued fraction expansions for the generating function of Euler polynomials, some of these formulae had been published by Stieltjes [14] and by Rogers [12] without proof. These formulae generalize our earlier results concerning Genocchi numbers, Euler numbers and Springer numbers [5, 4].

Euler polynomials Genocchi numbers continued fractions of Stieltjes-Rogers 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Références

  1. 1.
    D. André, “Développements de sec x et de tan x,” C.R. Acad. Sci. Paris 88 (1879), 965-967.Google Scholar
  2. 2.
    B. Berndt, Ramanujan's Notebook, Part II, Springer-Verlag, New York, 1985.Google Scholar
  3. 3.
    L. Comtet, Advanced Combinatorics, Dordrecht, Reidel, 1974.Google Scholar
  4. 4.
    D. Dumont, “Further triangles of Seidel-Arnold type and continued fractions related to Euler and Springer numbers,” Adv. Appl. Math. 16 (1995), 275-296.Google Scholar
  5. 5.
    D. Dumont and J. Zeng, “Further results on Euler and Genocchi numbers,” Aequationes Mathematics 47 (1994), 31-42.Google Scholar
  6. 6.
    P. Graham, D. Knuth, and O. Patashnik, Concrete Mathematics, 2nd edition, Addison-Wesley, 1994.Google Scholar
  7. 7.
    J.M. Hammersley, “An undergraduate exercise in manipulation,” The Mathematical Scientist 14 (1989), 1-23.Google Scholar
  8. 8.
    A.F. Horadam, “Generation of Genocchi polynomials of first order by recurrence relations,” Fibonacci Quart. 3 (1992), 239-243.Google Scholar
  9. 9.
    G. Kreweras, Communication privée, 1994.Google Scholar
  10. 10.
    C. Preece, “Theorems stated by Ramanujan (X),” J. London Math. Soc. 6 (1931), 23-32.Google Scholar
  11. 11.
    L. Rogers, “On the representation of certain asymptotic series as convergent continued fractions,” Proc. London Math. Soc. 4 (1906), 72-89.Google Scholar
  12. 12.
    L. Rogers, “Supplementary note on the representation of certain asymptotic series as convergent continued fractions,” Proc. London Math. Soc. 4 (1907), 393-395.Google Scholar
  13. 13.
    R. Stanley, Enumerative Combinatorics, Vol. 1, Wadsworth & Brooks/Cole, California, 1986.Google Scholar
  14. 14.
    T. Stieltjes, “Recherches sur les fractions continues,” Ann. Fac. Sci. Toulouse 9 (1895), 1-47.Google Scholar
  15. 15.
    G.X. Viennot, “Théorie combinatoire des nombres d'Euler et Genocchi,” Séminaire de Théorie des nombres, Exposé no. 11, Publications de l'Université de Bordeaux I, 1980–1981.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Dominique Dumont
    • 1
  • Jiang Zeng
    • 2
  1. 1.Département de MathématiqueUniversité Louis PasteurStrasbourg cédexFrance. Email
  2. 2.Institut Girard Desargues, MathématiquesUniversité Claude Bernard (Lyon I)Villeurbanne cédexFrance. Email

Personalised recommendations