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!}}{}_{3}F_{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 ${}_{3}F_{2}(\cdot )$ is the generalized hypergeometric functions

Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.

The dual Hahn polynomials have the orthogonality condition

\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 $n,m=0,1,...,N-1$ . 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 _{s=a}^{b-1}{\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.

Dual Hahn polynomials

Numerical instability

Relation to other polynomials

References

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