Faulhaber's formula, Bernoulli numbers, power sums of natural numbers and totatives and the functional equation $f(x)+x^k=f(x+1)$

Abstract

In modern usage the Bernoulli numbers and Bernoulli 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 that provide a characterization of Bernoulli numbers and related results. We then use these results to study sums of powers of totatives of $n$ that are less than $\frac{n}{2}$. In particular, we show that, like the case of the sum of powers of all totatives, the sum of powers of this half of the totatives can also be expressed as a linear combinations of Dirichlet inverses of Jordan totients of odd degrees.