On the coefficients of power sums of arithmetic progressions

We investigate the coefficients of the polynomial $$S_{m,r}^n(\mathrm{\ell })=r^n+(m+r{)}^n+(2m+r{)}^n+\cdots +((\mathrm{\ell }-1)m+r{)}^n\mathrm{.}$$ We prove that these can be given in terms of Stirling numbers of the first kind and $r$-Whitney numbers of the second kind. Moreover, we prove a necessary and sufficient condition for the integrity of these coefficients.