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
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
editThe dual Hahn polynomials have the orthogonality condition
for . Where ,
and
Numerical instability
editAs 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
for .
Then the orthogonality condition becomes
for
Relation to other polynomials
editThe 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
edit- 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.