identity involving translation for Bernoulli's polynomial

Question

Define the Bernoulli polynomials $B_k(x)$ by the power series expansion $$F(t,x) = \frac{te^{tx}}{e^t-1}=\sum_{k=0}^{\infty}B_k(x)\frac{t^k}{k!} \; .$$ Show that $$B_k(x)=N^{k-1}\sum_{a=0}^{N-1}B_k\left(\frac{x+a}{N}\right) \; .$$ I tried to do this by expand everything out but then ended up with some complicated-looking thing. Any suggestion or help is greatly appreciated.

Answer 1

Hint: First, work out that

$$\sum_{a=0}^{N-1}F\left(t,\frac{x+a}{N}\right)=NF\left(\frac{t}{N},x\right)$$