Cardano's formula, square roots, Chebyshev polynomials and radicals
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.
scientific article; zbMATH DE number 5651219