Given the following equality:
$$ B_n(sx) =s^{n - 1}\sum_{j = 0}^{s - 1} B_n (x + \frac{j}{s}) $$
where $B_n(x)$ - Bernoulli polynomial
How to prove the equality?
I tried to use generating function, but it has not brought any results. Maybe I choose wrong way.
Suppose we seek to verify that
$$B_n(qx) = q^{n-1} \sum_{j=0}^{q-1} B_n\left(x+\frac{j}{q}\right).$$
The EGF of the LHS is
$$\frac{t e^{qxt}}{e^t-1}.$$
The EGF of the RHS is
$$q^{n-1} \frac{t e^{xt}}{e^t-1} \sum_{j=0}^{q-1} e^{tj/q} = q^{n-1} \frac{t e^{xt}}{e^t-1} \frac{e^t-1}{e^{t/q}-1} \\ = q^{n-1} \frac{t e^{xt}}{e^{t/q}-1}.$$
The coefficients here are $$[t^n] q^{n-1} \frac{t e^{xt}}{e^{t/q}-1} = \frac{1}{q^n} [t^n] q^{n-1} \frac{tq e^{qxt}}{e^{t}-1} = [t^n] \frac{t e^{qxt}}{e^{t}-1}.$$
This is the same as the LHS, done.