Bernoulli polynomials and numbersBernoulliPolynomialsAndNumbers2008-04-07 17:13:542009-03-20 15:59:58DefinitionBernoulli polynomialBernoulli polynomialBernoulli number% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
% \usepackage[utf8]{inputenc}
\usepackage{amsthm}
\usepackage[T2A]{fontenc}
\usepackage[russian, english]{babel}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}For\, $n = 0,\,1,\,2,\,\ldots$,\, the {\em Bernoulli polynomial} may be defined as the uniquely determined polynomial $b_n(x)$ satisfying
\begin{align}
\int_x^{x+1}b_n(t)\,dt = x^n.
\end{align}
The constant term of $b_n(x)$ is the $n^{\mathrm{th}}$ {\em Bernoulli number} $B_n$.
The Bernoulli polynomial is often denoted also $B_n(x)$.\\
The uniqueness of the solution $b_n(x)$ in (1) is justificated by the
\textbf{Lemma.}\, For any polynomial $f(x)$, there exists a unique polynomial $g(x)$ with the same degree satisfying
\begin{align}
\int_x^{x+1}g(t)\,dt = f(x).
\end{align}
{\em Proof.}\, For every\, $n = 0,\,1,\,2,\,\ldots$,\, the polynomial
$$g_n(x) := \int_x^{x+1}t^n\,dt = \frac{(x+1)^{n+1}-x^{n+1}}{n+1}$$
is monic and its degree is $n$.\, If the coefficient of $x^n$ in $f(x)$ is $a_0$, then the difference $f(x)\!-\!a_0g_n(x)$ is a polynomial of degree $n\!-\!1$.\, Correspondingly we obtain $f(x)-a_0g_n(x)-a_1g_{n-1}(x)$ having the degree $n\!-\!2$ and so on.\, Finally we see that
$$f(x)-a_0g_n(x)-a_1g_{n-1}(x)-\ldots-a_ng_0(x)$$
must be the zero polynomial.\, Therefore
\begin{align*}
f(x) & = a_0g_n(x)+a_1g_{n-1}(x)+\ldots+a_ng_0(x)\\
& = \sum_{i=0}^na_ig_{n-i}(x)\\
& = \sum_{i=0}^na_i\int_x^{x+1}t^{n-i}\,dt\\
& = \int_x^{x+1}\sum_{i=0}^na_it^{n-i}\,dt
\end{align*}
whence we have\, $\displaystyle g(x) = \sum_{i=0}^na_ix^{n-i}$.\\
The proof implies also that the coefficients of $g(x)$ are rational, if the coefficients of $f(x)$ are such.\, So we know that all Bernoulli polynomials have only rational coefficients.
\begin{thebibliography}{7}
\bibitem{MMP} \CYRM. \CYRM. \CYRP\cyro\cyrs\cyrt\cyrn\cyri\cyrk\cyro\cyrv:
{\em \CYRV\cyrv\cyre\cyrd\cyre\cyrn\cyri\cyre\, \cyrv\, \cyrt\cyre\cyro\cyrr\cyri\cyryu\, \cyra\cyrl\cyrg\cyre\cyrb\cyrr\cyra\cyri\cyrch\cyre\cyrs\cyrk\cyri\cyrh \,
\cyrch\cyri\cyrs\cyre\cyrl}. \,\CYRI\cyrz\cyrd\cyra\cyrt\cyre\cyrl\cyrsftsn\cyrs\cyrt\cyrv\cyro \,
``\CYRN\cyra\cyru\cyrk\cyra''. \CYRM\cyro\cyrs\cyrk\cyrv\cyra \,(1982).
\end{thebibliography}
English translation:
M. M. Postnikov: \emph{Introduction to algebraic number theory}. Science Publs (``Nauka'').
Moscow (1982).