# Bernoulli polynomials and numbers

For  $n=0,\,1,\,2,\,\ldots$,  the 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 $n^{\mathrm{th}}$ $B_{n}$.

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  $n=0,\,1,\,2,\,\ldots$,  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 $f(x)\!-\!a_{0}g_{n}(x)$ 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

 $f(x)-a_{0}g_{n}(x)-a_{1}g_{n-1}(x)-\ldots-a_{n}g_{0}(x)$

must be the zero polynomial.  Therefore

 $\displaystyle f(x)$ $\displaystyle\;=\;a_{0}g_{n}(x)+a_{1}g_{n-1}(x)+\ldots+a_{n}g_{0}(x)$ $\displaystyle\;=\;\sum_{i=0}^{n}a_{i}g_{n-i}(x)$ $\displaystyle\;=\;\sum_{i=0}^{n}a_{i}\int_{x}^{x+1}t^{n-i}\,dt$ $\displaystyle\;=\;\int_{x}^{x+1}\sum_{i=0}^{n}a_{i}t^{n-i}\,dt$

whence we have  $\displaystyle g(x)=\sum_{i=0}^{n}a_{i}x^{n-i}$.

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.

## References

• 1 М. М. Постников: Введение  в  теорию  алгебраических  чисел.  Издательство  ‘‘Наука’’. Москва (1982).

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