Closed form for $\sum_{k=1}^n k^p$ through the Hermite integral representation of the Hurwitz zeta function

Abstract

Sums of positive integer powers have captivated the attention of mathematicians since ancient times. Over the centuries, mathematicians from diverse backgrounds have provided expressions for the sum of positive integer powers of the first $n$ positive integers. In this paper, we contribute to this endeavour by deriving Bernoulli's formula for $\sum_{k=1}^n k^p,$ for all positive integers $n$ and nonegative integers $p$, through the utilization of the Hermite integral representation of the Hurwitz zeta function.