Faulhaber's formula, Bernoulli numbers and the equation $f(x)+x^k=f(x+1)$

Abstract: In modern usage the Bernoulli numbers and polynomials follow Euler's approach and are defined using generating functions. We consider the functional equation $f(x)+x^k=f(x+1)$ and show that a solution can be derived from Faulhaber's formula for the sum of powers. We show how these solutions provide a characterization of Bernoulli numbers and related results.