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Transactions of the American Mathematical Society

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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 $\left | w \right | > \rho \quad (\rho > 0)$ 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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  • 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
  • Zeev Nehari, On the singularities of Legendre expansions, J. Rational Mech. Anal. 5 (1956), 987–992. MR 80747, DOI
  • 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

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Keywords: Jacobi polynomials, Faber polynomials
Article copyright: © Copyright 1990 American Mathematical Society