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
and are defined as
![{\displaystyle 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)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
for
and the parameters
are restricted to
.
Note that
is the rising factorial, otherwise known as the Pochhammer symbol, and
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
![{\displaystyle \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}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
for
. Where
,
![{\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)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
and
![{\displaystyle d_{n}^{2}={\frac {\Gamma (a+c+n+a)}{n!(b-a-n-1)!\Gamma (b-c-n)}}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Numerical instability
As the value of
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
![{\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}})]}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
for
.
Then the orthogonality condition becomes
![{\displaystyle \sum _{s=a}^{b-1}{\hat {w}}_{n}^{(c)}(s,a,b){\hat {w}}_{m}^{(c)}(s,a,b)=\delta _{m,n}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
for ![{\displaystyle n,m=0,1,...,N-1}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Relation to other polynomials
The Hahn polynomials,
, is defined on the uniform lattice
, and the parameters
are defined as
. Then setting
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.