On some generalizations of the sum of powers of natural numbers

TL;DR

This paper explores generalizations of the sum of powers of natural numbers, focusing on generating functions, binomial sums, and polynomial constructions for calculating these sums.

Contribution

It introduces new classes of sums with generating functions as powers of classical sums and provides methods to construct polynomials for their evaluation.

Findings

Analyzed sums with generating functions as powers of classical sums

Studied binomial sums and their properties

Developed polynomial methods for sum calculation

Abstract

In this paper some generalizations of the sum of powers of natural numbers is considered. In particular, the class of sums whose generating function is the power of the generating function for the classical sums of powers is studying. The so-called binomial sums are also considered. The problem of constructing polynomials that allow to calculate the values of the corresponding sums in certain cases is solved.