Bernoulli polynomials and numbers
The constant term of is the Bernoulli number .
The Bernoulli polynomial is often denoted also .
The uniqueness of the solution in (1) is justificated by the
Lemma. For any polynomial , there exists a unique polynomial with the same degree satisfying
(2) |
Proof. For every , the polynomial
is monic and its degree is . If the coefficient of in is , then the difference is a polynomial of degree . Correspondingly we obtain having the degree and so on. Finally we see that
must be the zero polynomial. Therefore
whence we have .
The proof implies also that the coefficients of are rational, if the coefficients of 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 |