§18.3 Definitions
…
► This table also includes the following special cases of Jacobi
polynomials :
ultraspherical , Chebyshev, and Legendre.
►
Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, normalizations, leading coefficients, and parameter constraints.
…
►
►
► For exact values of the coefficients of the Jacobi
polynomials
P
n
(
α
,
β
)
(
x
)
, the
ultraspherical polynomials
C
n
(
λ
)
(
x
)
, the Chebyshev
polynomials
T
n
(
x
)
and
U
n
(
x
)
, the Legendre
polynomials
P
n
(
x
)
, the Laguerre
polynomials
L
n
(
x
)
, and the Hermite
polynomials
H
n
(
x
)
, see
Abramowitz and Stegun (1964 , pp. 793–801) .
…The
ultraspherical polynomials are in powers of
x
for
n
=
0
,
1
,
…
,
6
.
…
…
► For Jacobi,
ultraspherical , Chebyshev, Legendre, and Hermite
polynomials , see Table
18.6.1 .
…
►
Table 18.6.1: Classical OP’s: symmetry and special values.
►
►
►
§18.6(ii) Limits to Monomials
…
►
18.6.4
lim
λ
→
∞
C
n
(
λ
)
(
x
)
C
n
(
λ
)
(
1
)
=
x
n
,
…
…
►
Ultraspherical (or Gegenbauer):
C
n
(
λ
)
(
x
)
.
…
►
Continuous
q
-Ultraspherical :
C
n
(
x
;
β
|
q
)
.
…
► In
Szegő (1975 , §4.7) the
ultraspherical polynomials
C
n
(
λ
)
(
x
)
are denoted by
P
n
(
λ
)
(
x
)
.
The
ultraspherical polynomials will not be considered for
λ
=
0
.
They are defined in the literature by
C
0
(
0
)
(
x
)
=
1
and
…
…
►
Table 18.5.1: Classical OP’s: Rodrigues formulas (18.5.5 ).
►
►
…
►
18.5.9
C
n
(
λ
)
(
x
)
=
(
2
λ
)
n
n
!
F
1
2
(
−
n
,
n
+
2
λ
λ
+
1
2
;
1
−
x
2
)
,
►
18.5.10
C
n
(
λ
)
(
x
)
=
∑
ℓ
=
0
⌊
n
/
2
⌋
(
−
1
)
ℓ
(
λ
)
n
−
ℓ
ℓ
!
(
n
−
2
ℓ
)
!
(
2
x
)
n
−
2
ℓ
=
(
2
x
)
n
(
λ
)
n
n
!
F
1
2
(
−
1
2
n
,
−
1
2
n
+
1
2
1
−
λ
−
n
;
1
x
2
)
,
►
18.5.11
C
n
(
λ
)
(
cos
θ
)
=
∑
ℓ
=
0
n
(
λ
)
ℓ
(
λ
)
n
−
ℓ
ℓ
!
(
n
−
ℓ
)
!
cos
(
(
n
−
2
ℓ
)
θ
)
=
e
i
n
θ
(
λ
)
n
n
!
F
1
2
(
−
n
,
λ
1
−
λ
−
n
;
e
−
2
i
θ
)
.
…
► Similarly in the cases of the
ultraspherical polynomials
C
n
(
λ
)
(
x
)
and the Laguerre
polynomials
L
n
(
α
)
(
x
)
we assume that
λ
>
−
1
2
,
λ
≠
0
, and
α
>
−
1
,
unless
stated otherwise .
…