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 and are defined as

for and the parameters are restricted to .

Note that is the falling and rising factorials, 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.


The Dual Hahn polynomials have the orthogonality condition

for . Where ,


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 renormalised Dual Hahn Polynomial as defined as

for .

Then the orthogonality condition becomes


Recurrence and difference relations

Rodrigues formula

Generating function

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 Tchebichef polynomials. Note that the Dual Hahn polynomials have a q-analog with an extra parameter q known as the Dual Hahn Q-polynomials

Racah polynomials are a generalization of dual Hahn polynomials


  • 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
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