# recurrence formula for Bernoulli numbers

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

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

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 RecurrenceFormulaForBernoulliNumbers 2013-03-22 17:46:19 2013-03-22 17:46:19 rm50 (10146) rm50 (10146) 4 rm50 (10146) Derivation msc 11B68