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}