A note on the hyper-sums of powers of integers, hyperharmonic polynomials and r-Stirling numbers of the first kind

TL;DR

This paper explores the relationships between hyper-sums of powers, hyperharmonic polynomials, and r-Stirling numbers, revealing new polynomial forms and identities involving Bernoulli and harmonic numbers.

Contribution

It establishes a polynomial form of r-Stirling numbers, links hyperharmonic polynomials to r-Stirling polynomials, and derives new identities involving classical number sequences.

Findings

Polynomial expression for r-Stirling numbers in terms of degree m-i

Connection between hyperharmonic polynomials and r-Stirling polynomials

New identities involving Bernoulli, Stirling, and harmonic numbers

Abstract

Recently, Kargin et al. (arXiv:2008.00284 [math.NT]) obtained (among many other things) the following formula for the hyper-sums of powers of integers $S_k^{(m)}(n)$ \begin{equation*} S_k^{(m)}(n) = \frac{1}{m!} \sum_{i=0}^{m} (-1)^i \genfrac{[}{]}{0pt}{}{m+n+1}{i+n+1}_{n+1} S_{k+i}(n), \end{equation*} where $S_k^{(0)}(n) \equiv S_k(n)$ is the ordinary power sum $1^k + 2^k + \cdots + n^k$. In this note we point out that a formula equivalent to the preceding one was already established in a different form, namely, a form in which $\genfrac{[}{]}{0pt}{}{m+n+1}{i+n+1}_{n+1}$ is given explicitly as a polynomial in $n$ of degree $m-i$. We find out the connection between this polynomial and the so-called $r$-Stirling polynomials of the first kind. Furthermore, we determine the hyperharmonic polynomials and their successive derivatives in terms of the $r$-Stirling polynomials of the first kind, and show the relationship between the (exponential) complete Bell polynomials and the $r$-Stirling numbers of the first kind. Finally, we derive some identities involving the Bernoulli numbers and polynomials, the $r$-Stirling numbers of the first kind, the Stirling numbers of both kinds, and the harmonic numbers.