Let B {\mathbf {B}} be an open bounded subset of the complex z z -plane with closure B ¯ \overline {\mathbf {B}} whose complement B ¯ c {\overline {\mathbf {B}} ^c} is a simply connected domain on the Riemann sphere. z = ψ ( w ) z = \psi (w) map the domain | w | > ρ ( ρ > 0 ) \left | w \right | > \rho \quad (\rho > 0) one-to-one conformally onto the domain B ¯ c {\overline {\mathbf {B}} ^c} such that ψ ( ∞ ) = ∞ \psi (\infty ) = \infty . Let R ( w ) = ∑ n = 0 ∞ c n w − n R(w) = \sum \nolimits _{n = 0}^\infty {{c_n}{w^{ - n}}} , c 0 ≠ 0 {c_0} \ne 0 be analytic in the domain | w | > ρ \left | w \right | > \rho with R ( w ) ≠ 0 R(w) \ne 0 . Let F ( z ) = ∑ n = 0 ∞ b n z n F(z) = \sum \nolimits _{n = 0}^\infty {{b_n}} {z^n} , F ∗ ( z ) = ∑ n = 0 ∞ 1 b n z n F*(z) = \sum \nolimits _{n = 0}^\infty {\frac {1} {{{b_n}}}} {z^n} be analytic in | z | > 1 \left | z \right | > 1 and analytically continuable to any point outside | z | > 1 \left | z \right | > 1 along any path not passing through the points z = 0 , 1 , ∞ z = 0,1,\infty . The generalized Faber polynomials { P n ( z ) } n = 0 ∞ \{ {P_n}(z)\} _{n = 0}^\infty of B {\mathbf {B}} are defined by \[ t ψ ′ ( t ) ψ ( t ) R ( t ) F ( z ψ ( t ) ) = ∑ n = 0 ∞ P n ( z ) 1 t n , | t | > ρ \frac {{t\psi ’(t)}} {{\psi (t)}}R(t)F\left ( {\frac {z} {{\psi (t)}}} \right ) = \sum \limits _{n = 0}^\infty {{P_n}(z)\frac {1} {{{t^n}}},} \quad \left | t \right | > \rho \] . The aim of this paper is to show that (1) if the Jacobi polynomials { P n ( α , β ) ( z ) } n = 0 ∞ \{ P_n^{(\alpha ,\beta )}(z)\} _{n = 0}^\infty are generalized Faber polynomials of any region B {\mathbf {B}} , then it must be the elliptic region { z : | z + 1 | + | z − 1 | > ρ + 1 ρ , ρ > 1 } ; \{ 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 ( α , α + 1 ) ( z ) } n = 0 ∞ \{ P_n^{(\alpha ,\alpha + 1)}(z)\} _{n = 0}^\infty , { P n ( β + 1 , β ) ( z ) } n = 0 ∞ \{ 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.