recurrence formula for Bernoulli numbers

This article establishes a well-known recurrence formula for the Bernoulli numbers.

The Bernoulli polynomials $b_{r}(x),r\geq 1$ can be written explicitly as

$b_{r}(x)=\sum_{k=1}^{r}\binom{r}{k}B_{r-k}x^{k}+B_{r}$

(see this article (http://planetmath.org/CoefficientsOfBernoulliPolynomials)).

For $r\geq 2$ , we have

$0=\int_{0}^{1}b_{r-1}(x)dx=\frac{1}{r}b_{r}(x)\big{\lvert}_{0}^{1}=\frac{1}{r}% (b_{r}(1)-b_{r}(0))$

and thus

$B_{r}=b_{r}(0)=b_{r}(1)=\sum_{k=1}^{r}\binom{r}{k}B_{r-k}+B_{r}$

It follows that (still when $r\geq 2$ )

$\sum_{k=1}^{r}\binom{r}{k}B_{r-k}=0$

so that

$\binom{r}{1}B_{r-1}=-\sum_{k=2}^{r}\binom{r}{k}B_{r-k}$

Replacing $r$ by $r+1$ and simplifying, we see that for $r\geq 1$ ,

$B_{r}=\frac{-1}{r+1}\sum_{k=2}^{r+1}\binom{r+1}{k}B_{r+1-k}=\frac{-1}{r+1}\sum% _{k=1}^{r}\binom{r+1}{k+1}B_{r-k}$

Title	recurrence formula for Bernoulli numbers
Canonical name	RecurrenceFormulaForBernoulliNumbers
Date of creation	2013-03-22 17:46:19
Last modified on	2013-03-22 17:46:19
Owner	rm50 (10146)
Last modified by	rm50 (10146)
Numerical id	4
Author	rm50 (10146)
Entry type	Derivation
Classification	msc 11B68