In mathematics, the dual Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined on a non-uniform lattice
x(s)=s(s+1)
(c) | ||
w | (s,a,b)= | |
n |
(a-b+1)n(a+c+1)n | |
n! |
{}3F2(-n,a-s,a+s+1;a-b+a,a+c+1;1)
n=0,1,...,N-1
a,b,c
- | 1 |
2 |
<a<b,|c|<1+a,b=a+N
Note that
(u)k
{}3F2( ⋅ )
give a detailed list of their properties.
The dual Hahn polynomials have the orthogonality condition
b-1 | |
\sum | |
s=a |
(c) | |
w | |
n |
(c) | |
(s,a,b)w | |
m |
(s,a,b)\rho(s)[\Deltax(s-
1 | |
2 |
)]=\deltanm
2 | |
d | |
n |
n,m=0,1,...,N-1
\Deltax(s)=x(s+1)-x(s)
\rho(s)= | \Gamma(a+s+1)\Gamma(c+s+1) |
\Gamma(s-a+1)\Gamma(b-s)\Gamma(b+s+1)\Gamma(s-c+1) |
| ||||
d | ||||
n |
.
As the value of
n
\hat
(c) | |
w | |
n |
(c) | ||
(s,a,b)=w | (s,a,b)\sqrt{ | |
n |
\rho(s) | ||||||
|
[\Deltax(s-
1 | |
2 |
)]}
n=0,1,...,N-1
Then the orthogonality condition becomes
b-1 | |
\sum | |
s=a |
\hat
(c) | |
w | |
n |
(s,a,b)\hat
(c) | |
w | |
m |
(s,a,b)=\deltam,n
n,m=0,1,...,N-1
The Hahn polynomials,
hn(x,N;\alpha,\beta)
x(s)=s
a,b,c
a=(\alpha+\beta)/2,b=a+N,c=(\beta-\alpha)/2
\alpha=\beta=0
Racah polynomials are a generalization of dual Hahn polynomials.