Recurrence formulas with gaps for Bernoulli and Euler polynomials

Abstract

Let $\{B_n\}$, $\{B_n(x)\}$ and $\{E_n(x)\}$ be the Bernoulli numbers, Bernoulli polynomials and Euler polynomials, respectively. In this paper we mainly establish formulas for ${\sum }_{6\mid k-3}\left(\genfrac{}{}{0ex}{}{n}{k}\right)B_{n-k}(x)$, ${\sum }_{6\mid k}\left(\genfrac{}{}{0ex}{}{n}{k}\right)E_{n-k}(x)$ and ${\sum }_{6\mid k-3}\left(\genfrac{}{}{0ex}{}{n}{k}\right)m^k{B}_{n-k}$ in the cases $m=2,3,4$.

arXiv comment16 pages