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- W2004754016 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 {{tpsi â(t)}} {{psi (t)}}R(t)Fleft ( {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." @default.
- W2004754016 created "2016-06-24" @default.
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- W2004754016 date "1990-01-01" @default.
- W2004754016 modified "2023-09-25" @default.
- W2004754016 title "Jacobi polynomials as generalized Faber polynomials" @default.
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- W2004754016 doi "https://doi.org/10.1090/s0002-9947-1990-0965745-1" @default.
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