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.

Orthogonality

The dual Hahn polynomials have the orthogonality condition

$\sum _{s=a}^{b-1}{w}_n^{(c)}(s,a,b)w_m^{(c)}(s,a,b)\rho (s)[\mathrm{\Delta }x(s-\frac{1}{2})]={\delta }_{nm}{d}_n^2$

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

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

and

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

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

${\hat{w}}_n^{(c)}(s,a,b)=w_n^{(c)}(s,a,b)\sqrt{\frac{\rho (s)}{d_n^2}[\mathrm{\Delta }x(s-\frac{1}{2})]}$

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.

References

• Zhu, Hongqing (2007), "Image analysis by discrete orthogonal dual Hahn moments", Pattern Recognition Letters 28 (13): 1688–1704, doi:10.1016/j.patrec.2007.04.013, https://www.hal.inserm.fr/inserm-00189813/file/Image_analysis_Hahn.pdf 
• Hahn, Wolfgang (1949), "Über Orthogonalpolynome, die q-Differenzengleichungen genügen", Mathematische Nachrichten 2 (1–2): 4–34, doi:10.1002/mana.19490020103, ISSN 0025-584X 
• Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8 
• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Hahn Class: Definitions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/18.19