A simple recursion for power sum polynomials (Q2509489)

The paper notes that the polynomial functions \(P_k(n)=\sum_{i=1}^n i^{k-1}\) satisfy the recursion \[ P_{k+1}(x)=k\int_0^x P_k(t)\,dt +x(1-k\int_0^1P_k(t)\,dt), \] with the initial condition \(P_1(x)=x\).