The Zagier modification of Bernoulli numbers and a polynomial extension. Part I - Archive ouverte HAL
Article Dans Une Revue Ramanujan Journal (The) Année : 2014

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

Atul Dixit
  • Fonction : Auteur
Victor H. Moll
  • Fonction : Auteur
Christophe Vignat

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 B2n+1(x) are classified. The strange 6-periodicity of B2n+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.

Dates et versions

hal-00931099 , version 1 (14-01-2014)

Identifiants

Citer

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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