coefficients of Bernoulli polynomials

The coefficient of xk in br(x) for k=1,2,,r is (rk)Br-k.

The proof is by inductionMathworldPlanetmath on r. For r=1, note that b1(x)=x-12, so that [x]b1(x)=1=(11)B0.

Writing [xk]f(x) for the coefficient of xk in a polynomial f(x), note that for k=1,2,,r,


since br(x)=rbr-1(x). By induction,


Thus the Bernoulli polynomialsMathworldPlanetmathPlanetmath can be written

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