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 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.
Orthogonality
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
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 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: