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arXiv · 1303.6590

The Zagier polynomials. Part II: Arithmetic properties of coefficients

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

The modified Bernoulli numbers \begin{equation*} B_{n}^{*} = \sum_{r=0}^{n} \binom{n+r}{2r} \frac{B_{r}}{n+r}, \quad n > 0 \end{equation*} introduced by D. 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. 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.

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Mark W. Coffey, Valerio De Angelis, Atul Dixit, Victor H. Moll, Armin Straub, Christophe Vignat. 2013-03-26. The Zagier polynomials. Part II: Arithmetic properties of coefficients. https://arxiv.org/abs/1303.6590

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