On the connection between a class of Faber polynomials and multivariable Chebyshev polynomials (Q2785335)

Let \(E\) be any compact set such that its complement with respect to the extended complex plane is simply connected. Then by the Riemann mapping theorem there is a conformal mapping \(w=\varphi (z)\) mapping the complement of \(E\) to the disk \(\{w\mid| w|> \rho_E\}\) where \(\rho_E\) is uniquely determined. Then the inverse function of \(\varphi\) can be written as \(z= \psi(w)= w+b_0+ {b_1 \over w}+ {b_2\over w^2} +\cdots\), and the polynomials \(F_n(z)\) in the relation \({\psi'(w) \over\psi(w)-z}= \sum^\infty_{n=0} {F_n(z) \over w^{n+1}}\) are called Faber polynomials. In the case that only finitely many \(b_k\) are nonzero, the author connects the Faber polynomials to the generalized Lucas polynomials and the Chebyshev polynomials of the first kind in several variables.