## 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.

### MSC:

 30C35 General theory of conformal mappings

### Keywords:

Faber polynomials; Caley-Hamilton theorem

Zbl 1010.33006
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### References:

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