Jacobi polynomials as generalized Faber polynomials
Authors: Zayed Ahmed I. 
Published: June 2012

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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction t psi prime left-parenthesis t right-parenthesis Over psi left-parenthesis t right-parenthesis EndFraction upper R left-parenthesis t right-parenthesis upper F left-parenthesis StartFraction z Over psi left-parenthesis t right-parenthesis EndFraction right-parenthesis equals sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts upper P Subscript n Baseline left-parenthesis z right-parenthesis StartFraction 1 Over t Superscript n Baseline EndFraction comma StartAbsoluteValue t EndAbsoluteValue greater-than rho">

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.