Bernoulli polynomials and numbers
For n=0, 1, 2,…, the Bernoulli polynomial may be defined as the uniquely determined polynomial
bn(x) satisfying
∫x+1xbn(t)𝑑t=xn. | (1) |
The constant term of bn(x) is the nth Bernoulli number 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
∫x+1xg(t)𝑑t=f(x). | (2) |
Proof. For every n=0, 1, 2,…, the polynomial
gn(x)=:∫x+1xtndt=(x+1)n+1-xn+1n+1 |
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
f(x)-a0gn(x)-a1gn-1(x)-…-ang0(x) |
must be the zero polynomial. Therefore
f(x) | =a0gn(x)+a1gn-1(x)+…+ang0(x) | ||
=n∑i=0aign-i(x) | |||
=n∑i=0ai∫x+1xtn-i𝑑t | |||
=∫x+1xn∑i=0aitn-idt |
whence we have g(x)=n∑i=0aixn-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 |