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

Bouali, Abdlilah

Bull. Sci. Math. 130, No. 1, 49-70 (2006).

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.

Reviewer: Wolfram Koepf (Kassel)

MSC:

30C35

General theory of conformal mappings

Keywords:

Faber polynomials; Caley-Hamilton theorem

Citations:

Zbl 1010.33006

References:

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