Jacobi polynomials as generalized Faber polynomials

Abstract:

Let ${\mathbf {B}}$ be an open bounded subset of the complex $z$-plane with closure $\overline {\mathbf {B}}$ whose complement ${\overline {\mathbf {B}} ^c}$ is a simply connected domain on the Riemann sphere. $z = \psi (w)$ map the domain $\left | w \right | > \rho \quad (\rho > 0)$ one-to-one conformally onto the domain ${\overline {\mathbf {B}} ^c}$ such that $\psi (\infty ) = \infty$. Let $R(w) = \sum \nolimits _{n = 0}^\infty {{c_n}{w^{ - n}}}$, ${c_0} \ne 0$ be analytic in the domain $\left | w \right | > \rho$ with $R(w) \ne 0$. Let $F(z) = \sum \nolimits _{n = 0}^\infty {{b_n}} {z^n}$, $F*(z) = \sum \nolimits _{n = 0}^\infty {\frac {1} {{{b_n}}}} {z^n}$ be analytic in $\left | z \right | < 1$ and analytically continuable to any point outside $\left | z \right | < 1$ along any path not passing through the points $z = 0,1,\infty$. The generalized Faber polynomials $\{ {P_n}(z)\} _{n = 0}^\infty$ of ${\mathbf {B}}$ are defined by \[ \frac {{t\psi ’(t)}} {{\psi (t)}}R(t)F\left ( {\frac {z} {{\psi (t)}}} \right ) = \sum \limits _{n = 0}^\infty {{P_n}(z)\frac {1} {{{t^n}}},} \quad \left | t \right | > \rho \]. The aim of this paper is to show that (1) if the Jacobi polynomials $\{ P_n^{(\alpha ,\beta )}(z)\} _{n = 0}^\infty$ are generalized Faber polynomials of any region ${\mathbf {B}}$, then it must be the elliptic region $\{ z:|z + 1| + |z - 1| < \rho + \frac {1}{\rho },\rho > 1\} ;$ (2) the only Jacobi polynomials that can be classified as generalized Faber polynomials are the Tchebycheff polynomials of the first kind, some normalized Gegenbauer polynomials, some normalized Jacobi polynomials of type $\{ P_n^{(\alpha ,\alpha + 1)}(z)\} _{n = 0}^\infty$, $\{ P_n^{(\beta + 1,\beta )}(z)\} _{n = 0}^\infty$ and there are no others, no matter how one normalizes them; (3) the Hermite and Laguerre polynomials cannot be generalized Faber polynomials of any region.