Advertisement

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
Article

Abstract

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.

Keywords

2-Adic valuations Digamma function Umbral calculus  Zagier polynomials 

Mathematics Subject Classification

Primary 11B68 11B83 

Notes

Acknowledgments

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.

References

  1. 1.
    Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York (1972)zbMATHGoogle Scholar
  2. 2.
    Alzer, H.: On some inequalities for the gamma and psi functions. Math. Comput. 66, 373–389 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Brychkov, Y.A.: Handbook of Special Functions. Derivatives, Integrals, Series and Other Formulas. Taylor and Francis, Boca Raton (2008)zbMATHGoogle Scholar
  4. 4.
    Carlitz, L.: A note on the Staudt–Clausen theorem. Am. Math. Mon. 64, 19–21 (1957)CrossRefzbMATHGoogle Scholar
  5. 5.
    Comtet, L.: Advanced Combinatorics. D. Reidel Publishing Co., Dordrecht (1974)CrossRefzbMATHGoogle Scholar
  6. 6.
    Dixit, A., Moll, V., Vignat, C.: The Zagier modification of Bernoulli numbers and a polynomial extension, Part I. Ramanujan J. 33, 379–422 (2014)Google 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.
    Guo, B.N., Qi, F.: Refinements of lower bounds for polygamma functions (2009). http://arxiv.org/abs/0903.1996v1
  9. 9.
    Guo, B.N., Qi, F.: Sharp inequalities for the psi function and harmonic numbers (2010). http://arxiv.org/abs/0902.2524
  10. 10.
    Guo, B.N., Chen, R.J., Qi, F.: A class of completely monotonic functions involving the polygamma functions. J. Math. Anal. Approx. Theory 1, 124–134 (2006)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Ireland, K., Rosen, M.: A Classical Introduction to Number Theory, 2nd edn. Springer, New York (1990)CrossRefzbMATHGoogle Scholar
  12. 12.
    Johnson, W.P.: The curious history of Faà di Bruno’s formula. Am. Math. Mon. 109, 217–234 (2002)CrossRefzbMATHGoogle Scholar
  13. 13.
    Olver, F.W.J.: Asymptotics and Special Functions. Academic Press, New York (1974).Google Scholar
  14. 14.
    Petkovšek, M., Wilf, H., Zeilberger, D.: A = B, 1st edn. AK Peters Ltd., Natick (1996)zbMATHGoogle Scholar
  15. 15.
    Riordan, J.: Combinatorial Identities, 1st edn. Wiley, New York (1968)zbMATHGoogle Scholar
  16. 16.
    Rogers, M.D.: Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46, 043509 (2005)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Zagier, D.: A modified Bernoulli number. Nieuw Archief voor Wiskunde 16, 63–72 (1998)zbMATHMathSciNetGoogle Scholar

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

Personalised recommendations