The Zagier polynomials. Part II: Arithmetic properties of coefficients

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• 作者：Mark W. Coffey (1)
  Valerio De Angelis (2)
  Atul Dixit (3)
  Victor H. Moll (3)
  Armin Straub (4)
  Christophe Vignat (3) (5)
  
• 关键词：2 ; Adic valuations ; Digamma function ; Umbral calculus ; Zagier polynomials ; Primary 11B68 ; 11B83
• 刊名：The Ramanujan Journal
• 出版年：2014
• 出版时间：December 2014
• 年：2014
• 卷：35
• 期：3
• 页码：361-390
• 全文大小：312 KB
• 参考文献：1. Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York (1972)
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  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)
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  15. Riordan, J.: Combinatorial Identities, 1st edn. Wiley, New York (1968)
  16. Rogers, M.D.: Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46, 043509 (2005) CrossRef
  17. Zagier, D.: A modified Bernoulli number. Nieuw Archief voor Wiskunde 16, 63鈥?2 (1998)
  
• 作者单位：Mark W. Coffey (1)
  Valerio De Angelis (2)
  Atul Dixit (3)
  Victor H. Moll (3)
  Armin Straub (4)
  Christophe Vignat (3) (5)
  
  1. Department of Physics, Colorado School of Mines, Golden, CO聽, 80401, USA
  2. Department of Mathematics, Xavier University of Louisiana, New Orleans, LA聽, 70125, USA
  3. Department of Mathematics, Tulane University, New Orleans, LA聽, 70118, USA
  4. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL聽, 61801, USA
  5. L.S.S. Supelec, Universite d鈥橭rsay, Orsay, France
  
• ISSN：1572-9303

文摘

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.