Advertisement

The Ramanujan Journal

, Volume 33, Issue 3, pp 379–422 | Cite as

The Zagier modification of Bernoulli numbers and a polynomial extension. Part I

  • Atul Dixit
  • Victor H. MollEmail author
  • Christophe Vignat
Article

Abstract

The modified Bernoulli numbers
$$ B_{n}^{*} = \sum_{r=0}^{n} \binom{n+r}{2r} \frac{B_{r}}{n+r}, \quad n > 0 $$
introduced by D. Zagier in 1998 are extended to the polynomial case by replacing B r by the Bernoulli polynomials B r (x). Properties of these new polynomials are established using the umbral method as well as classical techniques. The values of x that yield periodic subsequences \(B_{2n+1}^{*}(x)\) are classified. The strange 6-periodicity of \(B_{2n+1}^{*}\), established by Zagier, is explained by exhibiting a decomposition of this sequence as the sum of two parts with periods 2 and 3, respectively. Similar results for modifications of Euler numbers are stated.

Keywords

Bernoulli polynomials Chebyshev polynomials Umbral calculus Periodic sequences Euler polynomials Generating functions WZ-method 

Mathematics Subject Classification

11B68 33C45 

Notes

Acknowledgements

The authors wish to thank T. Amdeberhan for his valuable input into this paper.

References

  1. 1.
    Blissard, J.: Theory of generic equations. Pure Appl. Math. Q. 4, 279–305 (1861) Google Scholar
  2. 2.
    Boyadzhiev, K.: A note on Bernoulli polynomials and solitons. J. Nonlinear Math. Phys. 14, 174–178 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Brändén, P.: Iterated sequences and the geometry of zeros. J. Reine Angew. Math. 658, 115–131 (2011) zbMATHMathSciNetGoogle Scholar
  4. 4.
    Brychkov, Y.A.: Handbook of Special Functions. Derivatives, Integrals, Series and Other Formulas. Taylor and Francis, Boca Raton (2008) zbMATHGoogle Scholar
  5. 5.
    Coffey, M., Dixit, A., Moll, V., Straub, A., Vignat, C.: The Zagier modification of Bernoulli polynomials. Part II: arithmetic properties of denominators. Preprint (2012) Google Scholar
  6. 6.
    Gessel, I.: Applications of the classical umbral calculus. Algebra Univers. 49, 397–434 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Gradshteyn, I.S., Ryzhik, I.M.: In: Jeffrey, A., Zwillinger, D. (eds.) Table of Integrals, Series, and Products, 7th edn. Academic Press, New York (2007) Google Scholar
  8. 8.
    Grosset, M.P., Veselov, A.P.: Bernoulli numbers and solitons. J. Nonlinear Math. Phys. 12, 469–474 (2005) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Kervaire, M., Milnor, J.: Groups of homotopy spheres: I. Ann. Math. 77, 504–537 (1963) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Levine, J.P.: Lectures on groups of homotopy spheres. In: Ranicki, A., Levitt, N., Quinn, F. (eds.) Algebraic and Geometric Topology. Lecture Notes in Mathematics, vol. 1126, pp. 62–95. Springer, Berlin (1983) CrossRefGoogle Scholar
  11. 11.
    MacMillan, K., Sondow, J.: Proofs of power sum and binomial coefficient congruences via Pascal’s identity. Am. Math. Mon. 118, 549–551 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010) zbMATHGoogle Scholar
  13. 13.
    Petkovšek, M., Wilf, H., Zeilberger, D.: A=B, 1st edn. A.K. Peters, Wellesley (1996) Google Scholar
  14. 14.
    Ribenboim, P.: Fermat’s Last Theorem for Amateurs, 1st edn. Springer, New York (1999) zbMATHGoogle Scholar
  15. 15.
    Riordan, J.: Combinatorial Identities, 1st edn. Wiley, New York (1968) zbMATHGoogle Scholar
  16. 16.
    Spanier, J., Oldham, K.: An Atlas of Functions, 1st edn. Hemisphere, Washington (1987) zbMATHGoogle Scholar
  17. 17.
    Stewart, I., Tall, D.: Algebraic Number Theory, 1st edn. Chapman and Hall, London (1979) CrossRefzbMATHGoogle Scholar
  18. 18.
    Temme, N.M.: Special Functions. An Introduction to the Classical Functions of Mathematical Physics. Wiley, New York (1996) CrossRefzbMATHGoogle Scholar
  19. 19.
    Touchard, J.: Nombres exponentieles et nombres de Bernoulli. Can. J. Math. 8, 305–320 (1956) CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Zagier, D.: Hecke operators and periods of modular forms. Isr. Math. Conf. Proc. 3, 321–336 (1990) MathSciNetGoogle Scholar
  21. 21.
    Zagier, D.: A modified Bernoulli number. Nieuw Arch. Wiskd. 16, 63–72 (1998) zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Atul Dixit
    • 1
  • Victor H. Moll
    • 1
    Email author
  • Christophe Vignat
    • 1
    • 2
  1. 1.Department of MathematicsTulane UniversityNew OrleansUSA
  2. 2.L.S.S. SupelecUniversite d’OrsayOrsayFrance

Personalised recommendations