Skip to Main Content

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.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Jacobi polynomials as generalized Faber polynomials
HTML articles powered by AMS MathViewer

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


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, 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.
  • 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 10.1063/1.1704199
  • Zeev Nehari, On the singularities of Legendre expansions, J. Rational Mech. Anal. 5 (1956), 987–992. MR 80747, DOI 10.1512/iumj.1956.5.55038
  • V. J. Smirnov and N. Lebedev, 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.
  • 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 10.1080/17476938808814283
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 33A65, 30C20
  • Retrieve articles in all journals with MSC: 33A65, 30C20
Additional Information
  • © Copyright 1990 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 321 (1990), 363-378
  • MSC: Primary 33A65; Secondary 30C20
  • DOI:
  • MathSciNet review: 965745