Bernoulli polynomials and numbers

For  $n=0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots }$,  the Bernoulli polynomial may be defined as the uniquely determined polynomial $b_n(x)$ satisfying

${\displaystyle {\int }_x^{x+1}}{b}_n(t)𝑑t=x^n.$(1)

The constant term of $b_n(x)$ is the $n^{\mathrm{th}}$ 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

Lemma.  For any polynomial $f(x)$, there exists a unique polynomial $g(x)$ with the same degree satisfying

${\displaystyle {\int }_x^{x+1}}g(t)𝑑t=f(x).$(2)

Proof.  For every  $n=0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots }$,  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 $f(x)-a_0{g}_n(x)$ is a polynomial of degree $n-1$.  Correspondingly we obtain $f(x)-a_0{g}_n(x)-a_1{g}_{n-1}(x)$ having the degree $n-2$ and so on.  Finally we see that

$$f(x)-a_0{g}_n(x)-a_1{g}_{n-1}(x)-\mathrm{\dots }-a_n{g}_0(x)$$

must be the zero polynomial.  Therefore

$f(x)$$\mathrm{\hspace{0.33em}}=a_0{g}_n(x)+a_1{g}_{n-1}(x)+\mathrm{\dots }+a_n{g}_0(x)$

$\mathrm{\hspace{0.33em}}={\displaystyle \sum _{i=0}^n}{a}_i{g}_{n-i}(x)$

$\mathrm{\hspace{0.33em}}={\displaystyle \sum _{i=0}^n}{a}_i{\displaystyle {\int }_x^{x+1}}{t}^{n-i}𝑑t$

$\mathrm{\hspace{0.33em}}={\displaystyle {\int }_x^{x+1}}{\displaystyle \sum _{i=0}^n}{a}_i{t}^{n-i}dt$

whence we have  $g(x)={\displaystyle \sum _{i=0}^n}{a}_i{x}^{n-i}$.

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.

The relation (1) implies easily, that the Bernoulli polynomials form an Appell sequence.

English translation:

M. M. Postnikov: Introduction to algebraic number theory. Science Publs (‘‘Nauka’’).Moscow (1982).

Title	Bernoulli polynomials and numbers
Canonical name	BernoulliPolynomialsAndNumbers
Date of creation	2013-03-22 17:58:43
Last modified on	2013-03-22 17:58:43
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	13
Author	pahio (2872)
Entry type	Definition
Classification	msc 11B68
Synonym	Bernoulli numbers and polynomials
Related topic	BernoulliNumber
Related topic	CoefficientsOfBernoulliPolynomials
Related topic	TaylorSeriesViaDivision
Related topic	ReferenceRelatedToBernoulliPolynomialsAndNumbers
Related topic	EulerPolynomial
Defines	Bernoulli polynomial
Defines	Bernoulli number