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

Abstract

Recently, Kargın et al. (arXiv:2008.00284 [math.NT]) obtained (among many other things) the following formula for the hyper-sums of powers of integerswhereIn 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 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.