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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 caseby replacing $B_{r}$ by the Bernoulli polynomials $B_{r}(x)$. Properties ofthese new polynomials are established using the umbral method as well asclassical 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 thissequence as the sum of two parts with periods 2 and 3, respectively. Similarresults for modifications of Euler numbers are stated.

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