拉卡多項式 （Racah polynomials）是數學中以Guilio Racah命名的正交多項式，由下列廣義超幾何函數 定義[ 1]
拉卡多項式
p
n
(
x
(
x
+
γ
+
δ
+
1
)
)
=
4
F
3
[
−
n
n
+
α
+
β
+
1
−
x
x
+
γ
+
δ
+
1
α
+
1
γ
+
1
β
+
δ
+
1
;
1
]
.
{\displaystyle p_{n}(x(x+\gamma +\delta +1))={}_{4}F_{3}\left[{\begin{matrix}-n&n+\alpha +\beta +1&-x&x+\gamma +\delta +1\\\alpha +1&\gamma +1&\beta +\delta +1\\\end{matrix}};1\right].}
拉卡多項式的前數條是
h
y
p
e
r
g
e
o
m
(
[
−
1
,
−
x
,
2
+
a
+
b
,
x
+
c
+
d
+
1
]
,
[
a
+
1
,
c
+
1
,
b
+
d
+
1
]
,
1
)
h
y
p
e
r
g
e
o
m
(
[
−
2
,
−
x
,
3
+
a
+
b
,
x
+
c
+
d
+
1
]
,
[
a
+
1
,
c
+
1
,
b
+
d
+
1
]
,
1
)
h
y
p
e
r
g
e
o
m
(
[
−
3
,
−
x
,
4
+
a
+
b
,
x
+
c
+
d
+
1
]
,
[
a
+
1
,
c
+
1
,
b
+
d
+
1
]
,
1
)
h
y
p
e
r
g
e
o
m
(
[
−
4
,
−
x
,
5
+
a
+
b
,
x
+
c
+
d
+
1
]
,
[
a
+
1
,
c
+
1
,
b
+
d
+
1
]
,
1
)
h
y
p
e
r
g
e
o
m
(
[
−
5
,
−
x
,
6
+
a
+
b
,
x
+
c
+
d
+
1
]
,
[
a
+
1
,
c
+
1
,
b
+
d
+
1
]
,
1
)
h
y
p
e
r
g
e
o
m
(
[
−
6
,
−
x
,
7
+
a
+
b
,
x
+
c
+
d
+
1
]
,
[
a
+
1
,
c
+
1
,
b
+
d
+
1
]
,
1
)
.
{\displaystyle {\begin{aligned}hypergeom([-1,-x,2+a+b,x+c+d+1],[a+1,c+1,b+d+1],1)\\hypergeom([-2,-x,3+a+b,x+c+d+1],[a+1,c+1,b+d+1],1)\\hypergeom([-3,-x,4+a+b,x+c+d+1],[a+1,c+1,b+d+1],1)\\hypergeom([-4,-x,5+a+b,x+c+d+1],[a+1,c+1,b+d+1],1)\\hypergeom([-5,-x,6+a+b,x+c+d+1],[a+1,c+1,b+d+1],1)\\hypergeom([-6,-x,7+a+b,x+c+d+1],[a+1,c+1,b+d+1],1).\end{aligned}}}
拉卡多項式→哈恩多項式
lim
δ
→
∞
R
n
(
λ
(
x
)
;
−
N
−
1
,
δ
)
=
Q
n
(
x
;
α
,
β
,
N
)
{\displaystyle \lim _{\delta \to \infty }R_{n}(\lambda (x);-N-1,\delta )=Q_{n}(x;\alpha ,\beta ,N)}
拉卡多項式→雙重哈恩多項式
lim
β
→
∞
R
n
(
λ
(
x
)
;
−
N
−
1
,
β
,
γ
,
δ
)
=
R
n
(
λ
(
x
)
;
γ
,
δ
,
N
)
{\displaystyle \lim _{\beta \to \infty }R_{n}(\lambda (x);-N-1,\beta ,\gamma ,\delta )=R_{n}(\lambda (x);\gamma ,\delta ,N)}
![{\displaystyle p_{n}(x(x+\gamma +\delta +1))={}_{4}F_{3}\left[{\begin{matrix}-n&n+\alpha +\beta +1&-x&x+\gamma +\delta +1\\\alpha +1&\gamma +1&\beta +\delta +1\\\end{matrix}};1\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f8669e91de2c035849fc87d8ce46aa12600695e)
![{\displaystyle {\begin{aligned}hypergeom([-1,-x,2+a+b,x+c+d+1],[a+1,c+1,b+d+1],1)\\hypergeom([-2,-x,3+a+b,x+c+d+1],[a+1,c+1,b+d+1],1)\\hypergeom([-3,-x,4+a+b,x+c+d+1],[a+1,c+1,b+d+1],1)\\hypergeom([-4,-x,5+a+b,x+c+d+1],[a+1,c+1,b+d+1],1)\\hypergeom([-5,-x,6+a+b,x+c+d+1],[a+1,c+1,b+d+1],1)\\hypergeom([-6,-x,7+a+b,x+c+d+1],[a+1,c+1,b+d+1],1).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9f5bd393d24a60ce7d64dbb817a4463b8e8b971)
