双重哈恩多项式 (Dual Hahn polynomials)是一个正交多项式，定义如下[1]
Dual Hahn polynomials plot Dual Hahn polynomials plot
R
n
(
λ
(
x
)
;
γ
,
δ
,
N
)
=
3
F
2
(
−
n
,
−
x
,
x
+
γ
+
δ
+
1
;
γ
+
1
,
−
N
;
1
)
.
{\displaystyle R_{n}(\lambda (x);\gamma ,\delta ,N)={}_{3}F_{2}(-n,-x,x+\gamma +\delta +1;\gamma +1,-N;1).}
n ≤N 双重哈恩多项式的前几个：
双重哈恩多项式满足下列正交关系：[2]
∑
x
=
0
N
(
2
x
+
γ
+
δ
+
1
)
(
γ
+
1
)
x
(
−
N
)
x
N
!
(
−
1
)
x
(
x
+
γ
+
δ
+
1
)
N
+
1
(
δ
+
1
)
x
x
!
{\displaystyle \sum _{x=0}^{N}{\frac {(2x+\gamma +\delta +1)(\gamma +1)_{x}(-N)_{x}N!}{(-1)^{x}(x+\gamma +\delta +1)_{N+1}(\delta +1)_{x}x!}}}
R
m
(
λ
(
x
)
;
γ
,
δ
,
N
)
R
n
(
λ
(
x
)
;
γ
,
δ
,
N
)
=
δ
m
n
(
γ
+
n
n
)
(
δ
+
N
−
n
N
−
n
)
{\displaystyle R_{m}(\lambda (x);\gamma ,\delta ,N)R_{n}(\lambda (x);\gamma ,\delta ,N)={\frac {\delta _{mn}}{{\gamma +n \choose n}{\delta +N-n \choose N-n}}}}
极限关系
编辑
拉卡多项式 →双重哈恩多项式
lim
β
→
∞
R
n
(
λ
(
x
)
;
−
N
−
1
,
β
,
γ
,
δ
)
=
R
n
(
λ
(
lim
β
→
∞
R
n
(
λ
(
x
)
;
−
x
)
;
γ
,
δ
,
N
)
{\displaystyle \lim _{\beta \to \infty }R_{n}(\lambda (x);-N-1,\beta ,\gamma ,\delta )=R_{n}(\lambda (\lim _{\beta \to \infty }R_{n}(\lambda (x);-x);\gamma ,\delta ,N)}
双重哈恩多项式 →梅西纳多项式
lim
N
→
∞
R
n
(
λ
(
x
)
;
β
−
1
,
N
(
1
−
c
)
c
−
1
,
N
)
=
M
n
(
x
;
β
,
c
)
{\displaystyle \lim _{N\to \infty }R_{n}(\lambda (x);\beta -1,N(1-c)c^{-1},N)=M_{n}(x;\beta ,c)}
参考文献
编辑
^ Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York
^ KoeKoef p209
