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

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

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.

References

• Zhu, Hongqing (2007), "Image analysis by discrete orthogonal dual Hahn moments" (PDF), Pattern Recognition Letters, 28 (13): 1688–1704, doi:10.1016/j.patrec.2007.04.013
• Hahn, Wolfgang (1949), "Über Orthogonalpolynome, die q-Differenzengleichungen genügen", Mathematische Nachrichten, 2 (1–2): 4–34, doi:10.1002/mana.19490020103, ISSN 0025-584X, MR 0030647
• 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, MR 2656096
• 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.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.