Dual Hahn polynomials

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 $$x(s)=s(s+1)$$ and are defined as
 * $$w_n^{(c)} (s,a,b)=\frac{(a-b+1)_n(a+c+1)_n}{n!} {}_3F_2(-n,a-s,a+s+1;a-b+a,a+c+1;1)$$

for $$n=0,1,...,N-1$$ and the parameters $$a,b,c$$ are restricted to $$-\frac{1}{2}<a<b, |c|<1+a, b=a+N$$.

Note that $$(u)_k$$ is the rising factorial, otherwise known as the Pochhammer symbol, and $${}_3F_2(\cdot)$$ is the generalized hypergeometric functions

give a detailed list of their properties.

Orthogonality
The dual Hahn polynomials have the orthogonality condition
 * $$\sum^{b-1}_{s=a}w_n^{(c)}(s,a,b)w_m^{(c)}(s,a,b)\rho(s)[\Delta x(s-\frac{1}{2}) ]=\delta_{nm}d_n^2$$

for $$n,m=0,1,...,N-1$$. Where $$\Delta x(s)=x(s+1)-x(s)$$,
 * $$\rho(s)=\frac{\Gamma(a+s+1)\Gamma(c+s+1)}{\Gamma(s-a+1)\Gamma(b-s)\Gamma(b+s+1)\Gamma(s-c+1)}$$

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

Numerical instability
As the value of $$n$$ 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
 * $$\hat w_n^{(c)}(s,a,b)=w_n^{(c)}(s,a,b)\sqrt{\frac{\rho(s)}{d_n^2}[\Delta x(s-\frac{1}{2})]}$$

for $$n=0,1,...,N-1$$.

Then the orthogonality condition becomes
 * $$\sum^{b-1}_{s=a}\hat w_n^{(c)}(s,a,b)\hat w_m^{(c)}(s,a,b)=\delta_{m,n}$$

for $$n,m=0,1,...,N-1$$

Relation to other polynomials
The Hahn polynomials, $$h_n(x,N;\alpha,\beta)$$, is defined on the uniform lattice $$x(s)=s$$, and the parameters $$a,b,c$$ are defined as $$a=(\alpha+\beta)/2,b=a+N,c=(\beta-\alpha)/2$$. Then setting $$\alpha=\beta=0$$ 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.