The Zagier polynomials. Part II: Arithmetic properties of coefficients

TL;DR

This paper investigates the arithmetic properties of Zagier polynomials, focusing on the coefficients derived from modified Bernoulli numbers and polynomials, using various mathematical methods to analyze their properties.

Contribution

It extends the study of Zagier's modified Bernoulli numbers to polynomial cases and determines their 2-adic valuations and other arithmetic properties.

Findings

Arithmetic properties of Zagier polynomial coefficients established

2-adic valuation of modified Bernoulli numbers determined

Various analytic and asymptotic methods applied

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.