Several explicit formulae for Bernoulli polynomials

Abstract

We prove several explicit formulae for the $n$-th Bernoulli polynomial $B_{n}(x)$, in which $B_{n}(x)$ is equal to an affine combination of the polynomials $(x-1)^{n}$, $(x-2)^{n}$, $\ldots$, $(x-k-1)^{n}$, where $k$ is any fixed positive integer greater or equal than $n$.