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.