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Bernoulli polynomial

The Bernoulli polynomials are the sequence $\{b_{r}(x)\}_{r=0}^{\infty}$ of polynomials defined on $[0,1]$ by the conditions:

	$\displaystyle b_{0}(x)$	$\displaystyle=$	$\displaystyle 1,$
	$\displaystyle b^{\prime}_{r}(x)$	$\displaystyle=$	$\displaystyle rb_{r-1}(x),r\geq 1,$
	$\displaystyle\int_{0}^{1}b_{r}(x)dx$	$\displaystyle=$	$\displaystyle 0,r\geq 1$

These assumptions imply the identity

$\sum_{r=0}^{\infty}b_{r}(x)\frac{y^{r}}{r!}=\frac{ye^{xy}}{e^{y}-1}$

allowing us to calculate the $b_{r}$ . We have

	$\displaystyle b_{0}(x)$	$\displaystyle=$	$\displaystyle 1$
	$\displaystyle b_{1}(x)$	$\displaystyle=$	$\displaystyle x-\frac{1}{2}$
	$\displaystyle b_{2}(x)$	$\displaystyle=$	$\displaystyle x^{2}-x+\frac{1}{6}$
	$\displaystyle b_{3}(x)$	$\displaystyle=$	$\displaystyle x^{3}-\frac{3}{2}x^{2}+\frac{1}{2}x$
	$\displaystyle b_{4}(x)$	$\displaystyle=$	$\displaystyle x^{4}-2x^{3}+x^{2}-\frac{1}{30}$
	$\displaystyle\vdots$

Title	Bernoulli polynomial
Canonical name	BernoulliPolynomial
Date of creation	2013-03-22 11:45:51
Last modified on	2013-03-22 11:45:51
Owner	KimJ (5)
Last modified by	KimJ (5)
Numerical id	12
Author	KimJ (5)
Entry type	Definition
Classification	msc 11B68
Classification	msc 65-01
Related topic	BernoulliNumber