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
Canonical name	CoefficientsOfBernoulliPolynomials
Date of creation	2013-03-22 17:46:08
Last modified on	2013-03-22 17:46:08
Owner	rm50 (10146)
Last modified by	rm50 (10146)
Numerical id	4
Author	rm50 (10146)
Entry type	Derivation
Classification	msc 11B68
Related topic	BernoulliPolynomialsAndNumbers