Bernoulli polynomials and numbers

For  n=0, 1, 2,,  the Bernoulli polynomialMathworldPlanetmathPlanetmath may be defined as the uniquely determined polynomialPlanetmathPlanetmath bn(x) satisfying

xx+1bn(t)𝑑t=xn. (1)

The constant term of bn(x) is the nth Bernoulli numberMathworldPlanetmathPlanetmath Bn.

The Bernoulli polynomial is often denoted also Bn(x).

The uniqueness of the solution bn(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

xx+1g(t)𝑑t=f(x). (2)

Proof.  For every  n=0, 1, 2,,  the polynomial


is monic and its degree is n.  If the coefficient of xn in f(x) is a0, then the difference f(x)-a0gn(x) is a polynomial of degree n-1.  Correspondingly we obtain f(x)-a0gn(x)-a1gn-1(x) having the degree n-2 and so on.  Finally we see that


must be the zero polynomialPlanetmathPlanetmath.  Therefore

f(x) =a0gn(x)+a1gn-1(x)++ang0(x)

whence we have  g(x)=i=0naixn-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.


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

English translationMathworldPlanetmathPlanetmath:

M. M. Postnikov: Introduction to algebraic number theoryMathworldPlanetmath. 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