Continuous Hahn polynomials

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

p_n(x;a,b,c,d)= i^n\frac{(a+c)_n(a+d)_n}{n!}{}_3F_2(-n,n+a+b+c+d-1,a+ix;a+c,a+d;1)

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, 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.

Contents

Orthogonality

Recurrence and difference relations

Rodrigues formula

Generating function

Relation to other polynomials

References


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