Sums of powers and special polynomials

TL;DR

This paper explores sums of powers of positive integers, deriving their generating functions, expressing them via special polynomials, and connecting these to broader series and a problem posed by Ovidiu Furdui.

Contribution

It introduces explicit generating functions for sums of powers using exponential and geometric polynomials, linking them to existing series and a notable open problem.

Findings

Derived exponential and ordinary generating functions for sums of powers.

Expressed generating functions in terms of exponential and geometric polynomials.

Connected these series to an interesting problem of Ovidiu Furdui.

Abstract

In this paper, we discuss sums of powers of the positive integers and compute both the exponential and ordinary generating functions for these sums. We express these generating functions in terms of exponential and geometric polynomials and also show their connection to other interesting series. In particular, we show their connection to an interesting problem of Ovidiu Furdui.