Bernoulli Polynomials Question

Question

Could you please help me do this question. If I integrate the given expression for the derivative (2) to find $B_n(x)$, how can I find the value of the constant?

Answer 1

Call the function $G_n$ instead of $g$. First, observe $$G_n^\prime(x)=B_{n+1}^\prime(x+1)-B_{n+1}^\prime(x)=(n+1)(B_n(x+1)-B_n(x))$$

by the definition of the polynomials. Thus if we had proven $$B_n(x+1)-B_n(x)=nx^{n-1}$$ we would have $$G_n^\prime(x)=(n+1)nx^{n-1}$$ which means, integrating from $t=0$ to $t=x$, that $$\tag 1 G_n(x)-G_n(0)=(n+1)x^n$$

Note that $B_{n+1}^\prime=nB_n$ and $\displaystyle\int_0^1 B_n=0\;\;n\geqslant 1 $ gives by FTC that $$B_{n+1}(1)-B_{n+1}(0)=0\;\;n\geqslant 1$$

So, what is $G_n(0)$?