Faber polynomials, Cayley–Hamilton equation and Newton symmetric functions. (English) Zbl 1094.30010

Faber polynomials play an important role in different areas of mathematics. They are defined in the following way. Let \(K\) be a compact set in \(\mathbb{C}\), not a single point, whose complement \(\hat{\mathbb{C}}\setminus K\) (with respect to the extended plane) is simply connected. By the Riemann mapping theorem there exists a unique function \(z = \psi(w)\), meromorphic for \(| w| > 1\), which maps the domain \(| w| >1\) onto \(\hat{\mathbb{C}}\setminus K\) and satisfies the conditions \(\psi(\infty)=\infty, \psi (\infty) > 0.\) This condition implies that the function \(z = \psi (w)\), being analytic in the domain \(| w| >1\) without the point \(w = \infty\), has a simple pole at the point \(w = \infty\). The \(n\)th Faber polynomials of the first kind \(F_n(z)\) and of the second kind \(G_n(z)\) associated to \(\psi\) can be given from the following generating function \[ \frac{\psi(w)}{\psi(w)-z}= \sum \limits^\infty_{m\;=\;0}F_m(z)w^{-m-1}, \]
\[ \frac{1}{\psi(w)-z}= \sum\limits^\infty_{m\;=\;0}G_m(z)w^{-m-1}. \] The author gives an explicit formula for the Faber polynomials and for generalized Faber polynomials introduced byH. Airault and J. Ren in [“An algebra of differential operators and generating functions on the set of univalent functions”, Bull. Sci. Math. 126, No. 5, 343–367 (2002; Zbl 1010.33006)]. He introduces a new family of polynomials related to the Faber polynomials of the second kind. This allows him to give a generalized Cayley-Hamilton equation.


30C35 General theory of conformal mappings


Zbl 1010.33006
Full Text: DOI


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