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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Jacobi polynomials as generalized Faber polynomials


Author: Ahmed I. Zayed
Journal: Trans. Amer. Math. Soc. 321 (1990), 363-378
MSC: Primary 33A65; Secondary 30C20
MathSciNet review: 965745
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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\vert w \right\vert > \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\vert w \right\vert > \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\vert z \right\vert < 1$ and analytically continuable to any point outside $ \left\vert z \right\vert < 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

$\displaystyle \frac{{t\psi '(t)}} {{\psi (t)}}R(t)F\left( {\frac{z} {{\psi (t)}... ... 0}^\infty {{P_n}(z)\frac{1} {{{t^n}}},} \quad \left\vert t \right\vert > \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:\vert z + 1\vert + \vert z - 1\vert < \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.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1990-0965745-1
PII: S 0002-9947(1990)0965745-1
Keywords: Jacobi polynomials, Faber polynomials
Article copyright: © Copyright 1990 American Mathematical Society