Dual Hahn polynomials

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.

References