Faulhaber-type formulas for the sums of powers of arithmetic sequences

Abstract

In this article, we derive explicit formulas for computing the sums of powers in arithmetic sequences. We begin with a historical odyssey, tracing the contributions of some of the world’s most influential mathematicians whose work has shaped and inspired our approach. We then present two distinct Faulhaber-type formulas--one involving Bernoulli numbers and closely resembling the classical formula for sums of powers of integers. To establish these results, we employ two different techniques: the first is based on the principle of invariance, while the second uses the differencing operator applied to polynomials. Although the methods differ in form, we emphasize that they share the similar computational complexity, a point we demonstrate with illustrative examples at the end.