Sums of powers of integers and generalized Stirling numbers of the second kind

TL;DR

This paper derives new explicit formulas for sums of powers of integers using generalized Stirling numbers and the Newton-Gregory expansion, connecting these sums to Bernoulli polynomials and harmonic numbers.

Contribution

It introduces novel formulas for sums of powers involving generalized Stirling numbers of the second kind, expanding the mathematical tools available for these sums.

Findings

Derived infinite families of explicit formulas for $S_k(n)$

Connected sums of powers to generalized Stirling numbers and Bernoulli polynomials

Provided formulas involving dual and shifted Stirling numbers

Abstract

By applying the Newton-Gregory expansion to the polynomial associated with the sum of powers of integers $S_k(n) = 1^k + 2^k + \cdots + n^k$, we derive a couple of infinite families of explicit formulas for $S_k(n)$. One of the families involves the $r$-Stirling numbers of the second kind $\genfrac{\{}{\}}{0pt}{}{k}{j}_r$, $j=0,1,\ldots,k$, while the other involves their duals $\genfrac{\{}{\}}{0pt}{}{k}{j}_{-r}$, with both families of formulas being indexed by the non-negative integer $r$. As a by-product, we obtain three additional formulas for $S_k(n)$ involving the numbers $\genfrac{\{}{\}}{0pt}{}{k}{j}_{n+m}$, $\genfrac{\{}{\}}{0pt}{}{k}{j}_{n-m}$ (where $m$ is any given non-negative integer), and $\genfrac{\{}{\}}{0pt}{}{k}{j}_{k-j}$, respectively. Moreover, we provide a formula for the Bernoulli polynomials $B_k(x-1)$ in terms of $\genfrac{\{}{\}}{0pt}{}{k}{j}_{x}$ and the harmonic numbers.