The Zagier modification of Bernoulli numbers and a polynomial extension. Part I

TL;DR

This paper extends Zagier's modified Bernoulli numbers to Bernoulli polynomials, analyzing their properties, periodicity, and decompositions, and also discusses similar modifications for Euler numbers.

Contribution

It introduces a polynomial extension of Zagier's modified Bernoulli numbers and explores their properties, periodicity, and decompositions using classical and umbral methods.

Findings

Classification of x-values for periodic subsequences of B_{2n+1}^{*}(x)

Explanation of 6-periodicity via decomposition into periods 2 and 3

Extension of modifications to Euler numbers

Abstract

The modified B_{n}^{*} = \sum_{r=0}^{n} \binom{n+r}{2r} \frac{B_{r}}{n+r}, \quad n > 0 introduced by D. Zagier in 1998 are extended to the polynomial case by replacing $B_{r}$ by the Bernoulli polynomials $B_{r}(x)$. Properties of these new polynomials are established using the umbral method as well as classical techniques. The values of $x$ that yield periodic subsequences $B_{2n+1}^{*}(x)$ are classified. The strange 6-periodicity of $B_{2n+1}^{*}$, established by Zagier, is explained by exhibiting a decomposition of this sequence as the sum of two parts with periods 2 and 3, respectively. Similar results for modifications of Euler numbers are stated.