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

give a detailed list of their properties.

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

Orthogonality

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

In particular, they satisfy the orthogonality relation

for , , , , , .

Recurrence and difference relations

The sequence of continuous Hahn polynomials satisfies the recurrence relation

Rodrigues formula

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

Generating functions

The continuous Hahn polynomials have the following generating function:

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 F_n(x) are related to the special case a=b=c=d=1/2 of the continuous Hahn polynomials by

• The Jacobi polynomials P_n^{(α,β)}(x) can be obtained as a limiting case of the continuous Hahn polynomials:

References