# coefficients of Bernoulli polynomials

The coefficient of $x^{k}$ in $b_{r}(x)$ for $k=1,2,\ldots,r$ is $\binom{r}{k}B_{r-k}$.

The proof is by induction on $r$. For $r=1$, note that $b_{1}(x)=x-\frac{1}{2}$, so that $[x]b_{1}(x)=1=\binom{1}{1}B_{0}$.

Writing $[x^{k}]f(x)$ for the coefficient of $x^{k}$ in a polynomial $f(x)$, note that for $k=1,2,\ldots,r$,

 $[x^{k}]b_{r}(x)=\frac{1}{k}[x^{k-1}]b_{r}^{\prime}(x)=\frac{r}{k}[x^{k-1}]b_{r% -1}(x)$

since $b_{r}^{\prime}(x)=rb_{r-1}(x)$. By induction,

 $\frac{r}{k}[x^{k-1}]b_{r-1}(x)=\frac{r}{k}\binom{r-1}{k-1}B_{r-k}=\binom{r}{k}% B_{r-k}$

Thus the Bernoulli polynomials can be written

 $b_{r}(x)=\sum_{k=1}^{r}\binom{r}{k}B_{r-k}x^{k}+B_{r}$
Title coefficients of Bernoulli polynomials CoefficientsOfBernoulliPolynomials 2013-03-22 17:46:08 2013-03-22 17:46:08 rm50 (10146) rm50 (10146) 4 rm50 (10146) Derivation msc 11B68 BernoulliPolynomialsAndNumbers