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 



The authors wish to thank Larry Glasser for the proof given in Sect. 8, Karl Dilcher with help in the proof of Proposition 2.1, Christoph Koutschan for providing the expression for \(A_{n}\) given in Theorem 9.2, and Mathew Rogers for pointing out the result stated in Lemma 12.1. The authors also wish to thank Tewodros Amdeberhan for his valuable input into this paper.


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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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