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

@article{Dixit2012TheZM,
  title={The Zagier modification of Bernoulli numbers and a polynomial extension. Part I},
  author={Atul Abhay Dixit and Victor H. Moll and C. Vignat},
  journal={The Ramanujan Journal},
  year={2012},
  volume={33},
  pages={379-422}
}
AbstractThe modified Bernoulli numbers $$ B_{n}^{*} = \sum_{r=0}^{n} \binom{n+r}{2r} \frac{B_{r}}{n+r}, \quad n > 0 $$ 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… CONTINUE READING
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