連續雙哈恩多項式 (Continuous dual Hahn polynomials)是一個正交多項式，由下列廣義超幾何函數 定義[ 1]
連續雙哈恩多項式 連續雙哈恩多項式 複數圖
S
n
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x
2
;
a
,
b
,
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3
F
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i
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i
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a
+
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1
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{\displaystyle S_{n}(x^{2};a,b,c)={}_{3}F_{2}(-n,a+ix,a-ix;a+b,a+c;1).\ }
R
1
=
(
a
+
2.3
)
∗
(
a
+
3.5
)
∗
h
y
p
e
r
g
e
o
m
(
[
−
1
,
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+
I
∗
x
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a
−
I
∗
x
]
,
[
a
+
2.3
,
a
+
3.5
]
,
1
)
R
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=
p
o
c
h
h
a
m
m
e
r
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a
+
2.3
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2
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∗
p
o
c
h
h
a
m
m
e
r
(
a
+
3.5
,
2
)
∗
h
y
p
e
r
g
e
o
m
(
[
−
2
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a
+
I
∗
x
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a
−
I
∗
x
]
,
[
a
+
2.3
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a
+
3.5
]
,
1
)
R
3
=
p
o
c
h
h
a
m
m
e
r
(
a
+
2.3
,
3
)
∗
p
o
c
h
h
a
m
m
e
r
(
a
+
3.5
,
3
)
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h
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r
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o
m
(
[
−
3
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a
+
I
∗
x
,
a
−
I
∗
x
]
,
[
a
+
2.3
,
a
+
3.5
]
,
1
)
.
{\displaystyle {\begin{aligned}R1&=(a+2.3)*(a+3.5)*hypergeom([-1,a+I*x,a-I*x],[a+2.3,a+3.5],1)\\R2&=pochhammer(a+2.3,2)*pochhammer(a+3.5,2)*hypergeom([-2,a+I*x,a-I*x],[a+2.3,a+3.5],1)\\R3&=pochhammer(a+2.3,3)*pochhammer(a+3.5,3)*hypergeom([-3,a+I*x,a-I*x],[a+2.3,a+3.5],1).\end{aligned}}}
^ Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8 , MR 2656096
![{\displaystyle {\begin{aligned}R1&=(a+2.3)*(a+3.5)*hypergeom([-1,a+I*x,a-I*x],[a+2.3,a+3.5],1)\\R2&=pochhammer(a+2.3,2)*pochhammer(a+3.5,2)*hypergeom([-2,a+I*x,a-I*x],[a+2.3,a+3.5],1)\\R3&=pochhammer(a+2.3,3)*pochhammer(a+3.5,3)*hypergeom([-3,a+I*x,a-I*x],[a+2.3,a+3.5],1).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7e6b04ca8316d89d10932f6ca6ef11660bb8235)