Continuous dual Hahn polynomials

In mathematics, the continuous dual 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

${\displaystyle S_{n}(x^{2};a,b,c)={}_{3}F_{2}(-n,a+ix,a-ix;a+b,a+c;1).\ }$

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 R_n(x;γ,δ,N), the continuous Hahn polynomials p_n(x,a,b, a, b), and the Hahn polynomials. 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.