The Zagier modification of Bernoulli numbers and a polynomial extension. Part I
Résumé
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 Br by the Bernoulli polynomials Br(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.