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In matematica i polinomi di Jacobi costituiscono una sequenza polinomiale a due parametri e più precisamente costituiscono una successione di polinomi ortogonali a due parametri. Il loro nome ricorda il matematico tedesco Carl Jacobi (1804-1851).
Essi possono definirsi in molti modi equivalenti.
Mediante una serie ipergeometrica che in effetti si riduce a un polinomio:
P
n
(
α
,
β
)
(
z
)
:=
(
α
+
1
)
n
_
n
!
2
F
1
(
−
n
,
n
+
λ
,
α
+
1
;
1
−
z
2
)
,
{\displaystyle P_{n}^{(\alpha ,\beta )}(z):={\frac {(\alpha +1)^{\underline {n}}}{n!}}\,_{2}F_{1}\left(-n,n+\lambda ,\alpha +1;{\frac {1-z}{2}}\right),}
dove
n
_
{\displaystyle {\underline {n}}}
fattoriale crescente e dove
λ
:=
α
+
β
+
1
{\displaystyle \lambda :=\alpha +\beta +1}
Mediante la variante della precedente:
P
n
(
α
,
β
)
(
z
)
:=
(
−
1
)
n
(
β
+
1
)
n
_
n
!
2
F
1
(
−
n
,
n
+
λ
,
α
+
1
;
1
+
z
2
)
.
{\displaystyle P_{n}^{(\alpha ,\beta )}(z):={\frac {(-1)^{n}(\beta +1)^{\underline {n}}}{n!}}\,_{2}F_{1}\left(-n,n+\lambda ,\alpha +1;{\frac {1+z}{2}}\right).}
Mediante una formula alla Rodriguez :
P
n
(
α
,
β
)
(
z
)
:=
(
−
1
)
n
2
n
n
!
(
1
−
z
)
−
α
(
1
+
z
)
−
β
d
n
d
z
n
[
(
1
−
z
)
α
+
n
(
1
+
z
)
β
+
n
]
.
{\displaystyle P_{n}^{(\alpha ,\beta )}(z):={\frac {(-1)^{n}}{2^{n}n!}}\,(1-z)^{-\alpha }(1+z)^{-\beta }{\frac {d^{n}}{dz^{n}}}\left[(1-z)^{\alpha +n}(1+z)^{\beta +n}\right].}
Mediante la espressione polinomiale esplicita
P
n
(
α
,
β
)
(
z
)
:=
1
2
n
∑
k
=
0
n
(
n
+
α
k
)
(
n
+
β
n
−
k
)
(
z
−
1
)
n
−
k
(
z
+
1
)
k
.
{\displaystyle P_{n}^{(\alpha ,\beta )}(z):={\frac {1}{2^{n}}}\sum _{k=0}^{n}{n+\alpha \choose k}{n+\beta \choose n-k}(z-1)^{n-k}(z+1)^{k}.}
Come soluzioni polinomiali dell'equazione differenziale di Jacobi .
Per
α
,
β
>
−
1
{\displaystyle \alpha ,\beta >-1}
polinomi ortogonali nell'intervallo
[
−
1
,
1
]
{\displaystyle [-1,1]}
(
1
−
x
)
α
(
1
+
x
)
β
{\displaystyle (1-x)^{\alpha }(1+x)^{\beta }}
∫
−
1
1
(
1
−
x
)
α
(
1
+
x
)
β
P
m
α
,
β
(
x
)
P
n
α
,
β
(
x
)
d
x
=
{
0
,
se
m
≠
n
,
2
λ
Γ
(
n
+
α
+
1
)
Γ
(
n
+
β
+
1
)
(
2
n
+
λ
)
n
!
Γ
(
n
+
λ
)
,
se
m
=
n
≠
0
,
2
λ
Γ
(
α
+
1
)
Γ
(
β
+
1
)
Γ
(
λ
+
1
)
,
se
m
=
n
=
0.
{\displaystyle \int _{-1}^{1}(1-x)^{\alpha }(1+x)^{\beta }P_{m}^{\alpha ,\beta }(x)P_{n}^{\alpha ,\beta }(x)dx={\begin{cases}0,&{\text{se }}m\neq n,\\{\frac {2^{\lambda }\Gamma (n+\alpha +1)\Gamma (n+\beta +1)}{(2n+\lambda )n!\,\Gamma (n+\lambda )}},&{\text{se }}m=n\neq 0,\\{\frac {2^{\lambda }\Gamma (\alpha +1)\Gamma (\beta +1)}{\Gamma (\lambda +1)}},&{\text{se }}m=n=0.\end{cases}}}
Si tratta di varianti dei precedenti abbastanza modeste ma molto usate; sono definiti come
R
n
α
,
β
(
z
)
:=
P
n
(
α
,
β
)
(
2
z
−
1
)
.
{\displaystyle R_{n}^{\alpha ,\beta }(z):=P_{n}^{(\alpha ,\beta )}(2z-1).}
Naturalmente anche questi costituiscono una successione di polinomi ortogonali e la relazione di ortogonalità è:
∫
0
1
d
x
(
1
−
x
)
α
x
β
R
m
α
,
β
(
x
)
R
n
α
,
β
(
x
)
=
{
0
,
se
m
≠
n
,
Γ
(
n
+
α
+
1
)
Γ
(
n
+
β
+
1
)
(
2
n
+
λ
)
n
!
Γ
(
n
+
λ
)
,
se
m
=
n
≠
0
,
Γ
(
α
+
1
)
Γ
(
β
+
1
)
Γ
(
λ
+
1
)
,
se
m
=
n
=
0.
{\displaystyle \int _{0}^{1}dx(1-x)^{\alpha }x^{\beta }R_{m}^{\alpha ,\beta }(x)R_{n}^{\alpha ,\beta }(x)={\begin{cases}0,&{\text{se }}m\neq n,\\{\frac {\Gamma (n+\alpha +1)\Gamma (n+\beta +1)}{(2n+\lambda )n!\,\Gamma (n+\lambda )}},&{\text{se }}m=n\neq 0,\\{\frac {\Gamma (\alpha +1)\Gamma (\beta +1)}{\Gamma (\lambda +1)}},&{\text{se }}m=n=0.\end{cases}}}
Per
α
=
β
=
0
{\displaystyle \alpha =\beta =0}
polinomi di Legendre .
Per
α
=
β
{\displaystyle \alpha =\beta }
polinomi di Gegenbauer :
C
n
(
α
+
1
/
2
)
(
z
)
=
(
2
α
+
1
)
n
_
(
α
+
1
)
n
_
P
n
(
α
,
β
)
(
z
)
.
{\displaystyle C_{n}^{(\alpha +1/2)}(z)={\frac {(2\alpha +1)^{\underline {n}}}{\,}}{(\alpha +1){\underline {n}}}\,P_{n}^{(\alpha ,\beta )}(z).}
Per
α
=
β
=
−
1
/
2
{\displaystyle \alpha =\beta =-1/2}
polinomi di Chebyshev di primo genere:
T
n
(
z
)
=
n
!
(
1
/
2
)
n
_
P
n
(
−
1
/
2
,
−
1
/
2
)
(
z
)
.
{\displaystyle T_{n}(z)={\frac {n!}{(1/2)^{\underline {n}}}}P_{n}^{(-1/2,-1/2)}(z).}
I primi polinomi della successione graduale sono:
P
0
(
α
,
β
)
(
z
)
=
1
,
{\displaystyle P_{0}^{(\alpha ,\beta )}(z)=1,}
P
1
(
α
,
β
)
(
z
)
=
1
2
[
2
(
α
+
1
)
+
(
α
+
β
+
2
)
(
z
−
1
)
]
,
{\displaystyle P_{1}^{(\alpha ,\beta )}(z)={\frac {1}{2}}\left[2(\alpha +1)+(\alpha +\beta +2)(z-1)\right],}
P
2
(
α
,
β
)
(
z
)
=
1
8
[
4
(
α
+
1
)
(
α
+
2
)
+
4
(
α
+
β
+
3
)
(
α
+
2
)
(
z
−
1
)
+
(
α
+
β
+
3
)
(
α
+
β
+
4
)
(
z
−
1
)
2
]
.
{\displaystyle P_{2}^{(\alpha ,\beta )}(z)={\frac {1}{8}}\left[4(\alpha +1)(\alpha +2)+4(\alpha +\beta +3)(\alpha +2)(z-1)+(\alpha +\beta +3)(\alpha +\beta +4)(z-1)^{2}\right].}
![{\displaystyle P_{n}^{(\alpha ,\beta )}(z):={\frac {(-1)^{n}}{2^{n}n!}}\,(1-z)^{-\alpha }(1+z)^{-\beta }{\frac {d^{n}}{dz^{n}}}\left[(1-z)^{\alpha +n}(1+z)^{\beta +n}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c148cc297ff96ff838db8b92f32447dfddc4d87)
![{\displaystyle [-1,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51e3b7f14a6f70e614728c583409a0b9a8b9de01)
![{\displaystyle P_{1}^{(\alpha ,\beta )}(z)={\frac {1}{2}}\left[2(\alpha +1)+(\alpha +\beta +2)(z-1)\right],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d5189371212c709f50a42dd8b996959bca8de0e)
![{\displaystyle P_{2}^{(\alpha ,\beta )}(z)={\frac {1}{8}}\left[4(\alpha +1)(\alpha +2)+4(\alpha +\beta +3)(\alpha +2)(z-1)+(\alpha +\beta +3)(\alpha +\beta +4)(z-1)^{2}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a43f0a36b49b338a1438300ad2ef65c8db9e9889)
