A determinantal formula for the hyper-sums of powers of integers

Abstract

For non-negative integers r and m, let S (r) m (n) denote the r-fold summation (or hyper-sum) over the first n positive integers to the mth powers, with the initial condition S (0) m (n) = n m . In this paper, we derive a new determinantal formula for S

A note on another approach on power sums

“…Several methods to obtain explicit formulas for S (a) m (n) have been described in the literature (see, e.g., [1,2,3,4,6,7,8,9]). Recently, in a very interesting paper, C. Muschielok [10] has developed another approach on the iterated power sums S (a) m (n).…”