# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## Jacobi polynomials as generalized Faber polynomialsHTML articles powered by AMS MathViewer

by Ahmed I. Zayed
Trans. Amer. Math. Soc. 321 (1990), 363-378 Request permission

## 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.
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