Sums of powers of integers via differentiation

TL;DR

This paper introduces a differentiation-based method to derive recurrence relations and explicit formulas for sums of powers of integers, connecting them with Bernoulli numbers and Chebyshev polynomials.

Contribution

It develops novel recurrence relations for power sums using differentiation of generating functions and links these sums to Chebyshev polynomials and Bernoulli numbers.

Findings

Derived recurrence relations for even and odd power sums.

Established determinantal formulas for power sums and Bernoulli numbers.

Connected power sums to derivatives of Chebyshev polynomials.

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.