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 [math]\displaystyle{ x(s)=s(s+1) }[/math] and are defined as

[math]\displaystyle{ 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) }[/math]

for [math]\displaystyle{ n=0,1,...,N-1 }[/math] and the parameters [math]\displaystyle{ a,b,c }[/math] are restricted to [math]\displaystyle{ -\frac{1}{2}\lt a\lt b, |c|\lt 1+a, b=a+N }[/math].

Note that [math]\displaystyle{ (u)_k }[/math] is the rising factorial, otherwise known as the Pochhammer symbol, and [math]\displaystyle{ {}_3F_2(\cdot) }[/math] 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

[math]\displaystyle{ \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 }[/math]

for [math]\displaystyle{ n,m=0,1,...,N-1 }[/math]. Where [math]\displaystyle{ \Delta x(s)=x(s+1)-x(s) }[/math],

[math]\displaystyle{ \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)} }[/math]

and

[math]\displaystyle{ d_n^2=\frac{\Gamma(a+c+n+a)}{n!(b-a-n-1)!\Gamma(b-c-n)}. }[/math]

Numerical instability

As the value of [math]\displaystyle{ n }[/math] 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

[math]\displaystyle{ \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})]} }[/math]

for [math]\displaystyle{ n=0,1,...,N-1 }[/math].

Then the orthogonality condition becomes

[math]\displaystyle{ \sum^{b-1}_{s=a}\hat w_n^{(c)}(s,a,b)\hat w_m^{(c)}(s,a,b)=\delta_{m,n} }[/math]

for [math]\displaystyle{ n,m=0,1,...,N-1 }[/math]

Relation to other polynomials

The Hahn polynomials, [math]\displaystyle{ h_n(x,N;\alpha,\beta) }[/math], is defined on the uniform lattice [math]\displaystyle{ x(s)=s }[/math], and the parameters [math]\displaystyle{ a,b,c }[/math] are defined as [math]\displaystyle{ a=(\alpha+\beta)/2,b=a+N,c=(\beta-\alpha)/2 }[/math]. Then setting [math]\displaystyle{ \alpha=\beta=0 }[/math] 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