The Ramanujan Journal

, Volume 35, Issue 3, pp 361–390 | Cite as

The Zagier polynomials. Part II: Arithmetic properties of coefficients

  • Mark W. Coffey
  • Valerio De Angelis
  • Atul Dixit
  • Victor H. Moll
  • Armin Straub
  • Christophe Vignat


The modified Bernoulli numbers
$$\begin{aligned} B_{n}^{*} = \sum _{r=0}^{n} \left( {\begin{array}{c}n+r\\ 2r\end{array}}\right) \frac{B_{r}}{n+r}, \quad n > 0 \end{aligned}$$
introduced by Zagier in \(1998\) were recently extended to the polynomial case by replacing \(B_{r}\) by the Bernoulli polynomials \(B_{r}(x)\). Arithmetic properties of the coefficients of these polynomials are established here. In particular, the \(2\)-adic valuation of the modified Bernoulli numbers is determined. A variety of analytic, umbral, and asymptotic methods is used to analyze these polynomials.


2-Adic valuations Digamma function Umbral calculus  Zagier polynomials 

Mathematics Subject Classification

Primary 11B68 11B83 


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Mark W. Coffey
    • 1
  • Valerio De Angelis
    • 2
  • Atul Dixit
    • 3
  • Victor H. Moll
    • 3
  • Armin Straub
    • 4
  • Christophe Vignat
    • 3
    • 5
  1. 1.Department of PhysicsColorado School of MinesGoldenUSA
  2. 2.Department of MathematicsXavier University of LouisianaNew OrleansUSA
  3. 3.Department of MathematicsTulane UniversityNew OrleansUSA
  4. 4.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  5. 5.L.S.S. SupelecUniversite d’OrsayOrsayFrance

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