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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.07618 |
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Table of Contents:
- Let $P(m, X, N)$ be an $m$-degree polynomial in $X\in\mathbb{R}$ having fixed non-negative integers $m$ and $N$. The polynomial $P(m, X, N)$ is derived from a rearrangement of Faulhaber's formula in the context of Knuth's work entitled "Johann Faulhaber and sums of powers". In this manuscript we discuss the approximation properties of polynomial $P(m,X,N)$. In particular, the polynomial $P(m,X,N)$ approximates the odd power function $X^{2m+1}$ in a certain neighborhood of a fixed non-negative integer $N$ with a percentage error under $1\%$. By increasing the value of $N$ the length of convergence interval with odd-power $X^{2m+1}$ also increases. Furthermore, this approximation technique is generalized for arbitrary non-negative exponent $j$ of the power function $X^j$ by using splines.