Bernoulli polynomials and numbers
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
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.
- 1 М. М. Постников: Введение в теорию алгебраических чисел. Издательство ‘‘Наука’’. Москва (1982).
M. M. Postnikov: Introduction to algebraic number theory. Science Publs (‘‘Nauka’’). Moscow (1982).
|Title||Bernoulli polynomials and numbers|
|Date of creation||2013-03-22 17:58:43|
|Last modified on||2013-03-22 17:58:43|
|Last modified by||pahio (2872)|
|Synonym||Bernoulli numbers and polynomials|