Obtaining Easily Powers Sums on Arithmetic Progressions and Properties of Bernoulli Polynomials by Operator Calculus

We show that a sum of powers on an arithmetic progression is the transform of a monomial by a
differential operator and that its generating function is simply related to that of the Bernoulli
polynomials from which consequently it may be calculated. Besides, we show that it is obtainable also
from the sums of powers of integers, i.e. from the Bernoulli numbers which in turn may be calculated
by a simple algorithm.
By the way, for didactic purpose, operator calculus is utilized for proving in a concise manner the main
properties of the Bernoulli polynomials.