Evaluation of Some Sums Involving Powers of Harmonic Numbers

In this note, we extend the definition of multiple harmonic sums and apply their stuffle relations to obtain explicit evaluations of the sums $R_n(p,t)=sum olimits_{m=0}^n m^p H_m^t$, where $H_m$ are harmonic numbers. When $tle 4$ these sums were first studied by Spiess around 1990 and, more recently, by Jin and Sun. Our key step first is to find an explicit formula of a special type of the extended multiple harmonic sums. This also enables us to provide a general structural result of the sums $R_n(p,t)$ for all $tge 0$.