Dual Hahn polynomials

The dual Hahn polynomials have the orthogonality condition

${\displaystyle \sum _{s=a}^{b-1}w_{n}^{(c)}(s,a,b)w_{m}^{(c)}(s,a,b)\rho (s)[\Delta x(s-{\frac {1}{2}})]=\delta _{nm}d_{n}^{2}}$

for ${\displaystyle n,m=0,1,...,N-1}$. Where ${\displaystyle \Delta x(s)=x(s+1)-x(s)}$,

${\displaystyle \rho (s)={\frac {\Gamma (a+s+1)\Gamma (c+s+1)}{\Gamma (s-a+1)\Gamma (b-s)\Gamma (b+s+1)\Gamma (s-c+1)}}}$

and

${\displaystyle d_{n}^{2}={\frac {\Gamma (a+c+n+a)}{n!(b-a-n-1)!\Gamma (b-c-n)}}.}$

As the value of ${\displaystyle n}$ increases, the values that the discrete polynomials obtain also increases. As a result, to obtain numerical stability in calculating the polynomials you would use the renormalized dual Hahn polynomial as defined as

${\displaystyle {\hat {w}}_{n}^{(c)}(s,a,b)=w_{n}^{(c)}(s,a,b){\sqrt {{\frac {\rho (s)}{d_{n}^{2}}}[\Delta x(s-{\frac {1}{2}})]}}}$

for ${\displaystyle n=0,1,...,N-1}$.

Then the orthogonality condition becomes

${\displaystyle \sum _{s=a}^{b-1}{\hat {w}}_{n}^{(c)}(s,a,b){\hat {w}}_{m}^{(c)}(s,a,b)=\delta _{m,n}}$

for ${\displaystyle n,m=0,1,...,N-1}$

The Hahn polynomials, ${\displaystyle h_{n}(x,N;\alpha ,\beta )}$, is defined on the uniform lattice ${\displaystyle x(s)=s}$, and the parameters ${\displaystyle a,b,c}$ are defined as ${\displaystyle a=(\alpha +\beta )/2,b=a+N,c=(\beta -\alpha )/2}$. Then setting ${\displaystyle \alpha =\beta =0}$ the Hahn polynomials become the Chebyshev polynomials. Note that the dual Hahn polynomials have a q-analog with an extra parameter q known as the dual q-Hahn polynomials.

Racah polynomials are a generalization of dual Hahn polynomials.

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