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Recently, Kargin et al. (arXiv:2008.00284 [math.NT]) obtained (among manyother 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 thepreceding one was already established in a different form, namely, a form inwhich $\genfrac{[}{]}{0pt}{}{m+n+1}{i+n+1}_{n+1}$ is given explicitly as apolynomial in $n$ of degree $m-i$. We find out the connection between thispolynomial and the so-called $r$-Stirling polynomials of the first kind.Furthermore, we determine the hyperharmonic polynomials and their successivederivatives in terms of the $r$-Stirling polynomials of the first kind, andshow the relationship between the (exponential) complete Bell polynomials andthe $r$-Stirling numbers of the first kind. Finally, we derive some identitiesinvolving the Bernoulli numbers and polynomials, the $r$-Stirling numbers ofthe first kind, the Stirling numbers of both kinds, and the harmonic numbers.

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