Uma fórmula geral não recursiva da soma das funções polinomiais dos inteiros positivos

The main objective of this work is to present a non-recursive general formula that determines the sum of polynomial functions applied to positive integers. For this, some generalizations of the concept of functions and sequences using the higher-order difference operator are presented. Thus, with this, we obtain a formula for the sum of the powers of non-negative integers, which is a particular case of polynomial functions. Then, a formula for determining the general term of a higher order arithmetic progression is obtained, as well as a formula for determining the sum of the first n terms of it. And finally, you get a formula to determine a figured number and the sum of them for whatever the number of sides