Several explicit formulae for Bernoulli polynomials

Takao Komatsu; Claudio de J. Pita Ruiz V.

Vol. 21 No. 1 (2016)

Regular Articles

Several explicit formulae for Bernoulli polynomials

Published 2016-02-17

Takao Komatsu
Claudio de J. Pita Ruiz V.

Submission of an original manuscript to the Mathematical Communications will be taken to mean that it represents original work not previously published, that it is not being considered elsewhere for publication; and, if accepted for publication, it will be published in print and online form and it will not be published elsewhere in the same form, for commercial purposes, in any language, without the consent of the publisher. The journal takes the stance that the publication of scholarly research is meant to disseminate knowledge and, in a not-for-profit regime, benefits neither publisher nor author financially. It sees itself as having an obligation to its authors and to society to make content available online now that the technology allows for such a possibility.

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 $n$.

Keywords

Bernoulli polynomials, binomial coefficients, Bernoulli numbers

PDF

Supplementary File(s)

LaTeX file