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The Zagier polynomials. Part II: arithmetic properties of coefficients

The Zagier polynomials. Part II: arithmetic properties of coefficients
Mark W. Coffey, Valerio De Angelis, Atul Dixit, Victor H. Moll, Armin Straub, Christophe VignatThe Ramanujan Journal — Volume 35, Number 3, 2014, Pages 361-390

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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BibTeX

@article{bernoullizagier-2014,
author = {Mark W. Coffey and Valerio De Angelis and Atul Dixit and Victor H. Moll and Armin Straub and Christophe Vignat},
title = {The {Z}agier polynomials. {Part II}: arithmetic properties of coefficients},
journal = {The Ramanujan Journal},
year = {2014},
volume = {35},
number = {3},
pages = {361--390},
doi = {10.1007/s11139-014-9568-5},
}