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

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.
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Submitted on : Tuesday, January 14, 2014 - 8:34:21 PM
Last modification on : Thursday, April 5, 2018 - 12:30:04 PM

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Atul Dixit, Victor H. Moll, Christophe Vignat. The Zagier modification of Bernoulli numbers and a polynomial extension. Part I. Ramanujan Journal (The), 2014, 33 (3), pp.379-422. ⟨10.1007/s11139-013-9484-0⟩. ⟨hal-00931099⟩

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