Cardano's formula, square roots, Chebyshev polynomials and radicals
type: Item
instance of: scholarly article
title: Cardano's formula, square roots, Chebyshev polynomials and radicals
zbMATH Open document ID: 1221.12007
author: Roman Wituła
author: Damian Słota
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-12-22T00:00:00Z
review text: Let \(p,q\in{\mathbb C}\), \(pq\neq 0\), \(q^2\neq 4p^n\), \(n\in{\mathbb N}\). The authors prove: \[ (\sqrt{p})^n\Omega_n\left(\frac{x}{\sqrt{p}}\right)-q=\prod_{k=0}^{n-1}(x-a\zeta^{2k}-b\zeta^{-2k}) \] and \[ (-\sqrt{-p})^n\Omega_n\left(\frac{ix}{\sqrt{p}}\right)+q=\prod_{k=0}^{n-1}(x-a\zeta^{2k+1}-b\zeta^{-2k-1}), \] where \(\Omega_n(x)=2T_n\left(\frac{x}{2}\right)\), \(T_n\) the \(n\)th Chebyshev polynomial of the first kind, \(\zeta=\exp\left(\frac{i\pi}{n}\right)\), \(a\) and \(b\) are one of \[ \sqrt[n]{\frac12 (q+\sqrt{q^2-4p^n})}, \sqrt[n]{\frac12 (q-\sqrt{q^2-4p^n})}, \] respectively, \(ab=p\). Formulas for the roots of Vietà-Fibonacci functions are also presented, together with applications to generating radical-type identities.
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 12E10
Mathematics Subject Classification ID: 11C08
Mathematics Subject Classification ID: 12E12
zbMATH DE Number: 5651219
zbMATH Keywords: Cardano's formula
zbMATH Keywords: square roots
zbMATH Keywords: Chebyshev polynomials
zbMATH Keywords: radicals
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.09.056
OpenAlex ID: W2040051652
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DOI: 10.1016/J.JMAA.2009.09.056
description: scientific article; zbMATH DE number 5651219