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


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.


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

Mathematics Subject Classification

11B68 33C45 



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


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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

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