Jacobi polynomials as generalized Faber polynomials

Author:
Ahmed I. Zayed

Journal:
Trans. Amer. Math. Soc. **321** (1990), 363-378

MSC:
Primary 33A65; Secondary 30C20

DOI:
https://doi.org/10.1090/S0002-9947-1990-0965745-1

MathSciNet review:
965745

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

- Richard Askey,
*Orthogonal polynomials and special functions*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975. MR**0481145**
G. Faber, - R. P. Gilbert,
*Integral operator methods in bi-axially symmetric potential theory*, Contributions to Differential Equations**2**(1963), 441–456 (1963). MR**156998** - R. P. Gilbert,
*Bergman’s integral operator method in generalized axially symmetric potential theory*, J. Mathematical Phys.**5**(1964), 983–997. MR**165131**, DOI https://doi.org/10.1063/1.1704199 - Zeev Nehari,
*On the singularities of Legendre expansions*, J. Rational Mech. Anal.**5**(1956), 987–992. MR**80747**, DOI https://doi.org/10.1512/iumj.1956.5.55038
V. J. Smirnov and N. Lebedev, - P. K. Suetin,
*The basic porperties of Faber polynomials*, Uspehi Mat. Nauk**19**(1964), no. 4 (118), 125–154 (Russian). MR**0168773** - P. K. Suetin,
*Fundamental properties of polynomials orthogonal on a contour*, Uspehi Mat. Nauk**21**(1966), no. 2 (128), 41–88 (Russian). MR**0198111** - P. K. Suetin,
*Mnogochleny, ortogonal′nye po ploshchadi, i mnogochleny Bieberbakha*, Izdat. “Nauka”, Moscow, 1971 (Russian). Trudy Mat. Inst. Steklov. 100 (1971). MR**0463792**
G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1975.
- A. Zayed, M. Freund, and E. Görlich,
*A theorem of Nehari revisited*, Complex Variables Theory Appl.**10**(1988), no. 1, 11–22. MR**946095**, DOI https://doi.org/10.1080/17476938808814283

*Über polynomische Entwicklungen*, Math. Ann.

**57**(1903), 389-408;

**64**(1907), 116-135. Ya. L. Geronimus,

*Polynomials, orthogonal on a a circle and on an interval*, Fizmatgiz, Moscow, 1958.

*Functions of a complex variable*, M. I. T. Press, Cambridge, Mass., 1968. H. Srivatstave and H. Manocha,

*A treatise on generating functions*, Ellis Horwood, West Sussex, England, 1984.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
33A65,
30C20

Retrieve articles in all journals with MSC: 33A65, 30C20

Additional Information

Keywords:
Jacobi polynomials,
Faber polynomials

Article copyright:
© Copyright 1990
American Mathematical Society