Continuous Hahn polynomials

In mathematics, the continuous Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined in terms of generalized hypergeometric functions by

Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010,14) give a detailed list of their properties.

Closely related polynomials include the dual Hahn polynomials Rn(x;γ,δ,N), the Hahn polynomials Qn(x;a,b,c), and the continuous dual Hahn polynomials Sn(x;a,b,c). These polynomials all have q-analogs with an extra parameter q, such as the q-Hahn polynomials Qn(x;α,β, N;q), and so on.


The continuous Hahn polynomials pn(x;a,b,c,d) are orthogonal with respect to the weight function

In particular, they satisfy the orthogonality relation[1][2][3]

for , , , , , .

Recurrence and difference relations

The sequence of continuous Hahn polynomials satisfies the recurrence relation[4]

Rodrigues formula

The continuous Hahn polynomials are given by the Rodrigues-like formula[5]

Generating functions

The continuous Hahn polynomials have the following generating function:[6]

A second, distinct generating function is given by

Relation to other polynomials

  • The Wilson polynomials are a generalization of the continuous Hahn polynomials.
  • The Bateman polynomials Fn(x) are related to the special case a=b=c=d=1/2 of the continuous Hahn polynomials by
  • The Jacobi polynomials Pn(α,β)(x) can be obtained as a limiting case of the continuous Hahn polynomials:[7]


  1. Koekoek, Lesky, & Swarttouw (2010), p. 200.
  2. Askey, R. (1985), "Continuous Hahn polynomials", J. Phys. A: Math. Gen. 18: pp. L1017-L1019.
  3. Andrews, Askey, & Roy (1999), p. 333.
  4. Koekoek, Lesky, & Swarttouw (2010), p. 201.
  5. Koekoek, Lesky, & Swarttouw (2010), p. 202.
  6. Koekoek, Lesky, & Swarttouw (2010), p. 202.
  7. Koekoek, Lesky, & Swarttouw (2010), p. 203.
  • Hahn, Wolfgang (1949), "Über Orthogonalpolynome, die q-Differenzengleichungen genügen", Mathematische Nachrichten, 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
  • Andrews, George E.; Askey, Richard; Roy, Ranjan (1999), Special functions, Encyclopedia of Mathematics and its Applications 71, Cambridge: Cambridge University Press, ISBN 978-0-521-62321-6
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