Sums of powers of integers via differentiation

Abstract

For integer $k \geq 0$, let $S_k$ denote the sum of the $k$th powers of the first $n$ positive integers $1^k + 2^k + \cdots + n^k$. For any given $k$, the power sum $S_k$ can in principle be determined by differentiating $k$ times (with respect to $x$) the associated exponential generating function $\sum_{k=0}^{\infty}S_k x^k/k!$, and then taking the limit of the resulting differentiated function as $x$ approaches $0$. In this paper, we exploit this method to establish a couple of seemingly novel recurrence relations, one of them involving the even-indexed power sums $S_2, S_4,\ldots, S_{2k}$, and the other the odd-indexed power sums $S_{1}, S_3, \ldots, S_{2k-1}$, with both recurrence relations depending explicitly on the parameter $N = n + \frac{1}{2}$. From this, we obtain a determinantal formula of order $k$ which yields $S_{2k}$ [$S_{2k-1}$] in the Faulhaber form, that is, as an odd [even] polynomial in $N$. As a byproduct, we discover a new determinantal formula for the Bernoulli number $B_{2k}$. Furthermore, we show that $S_{2k}$ and $S_{2k-1}$ can be obtained by taking the corresponding higher order derivatives of the Chebyshev polynomials of the second kind.