In mathematics, the 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
for 0≤n≤N where λ(x)=x(x+γ+δ+1).
Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.
Closely related polynomials include the Hahn polynomials, the continuous Hahn polynomials pn(x,a,b, a, b), 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.
Relation to other polynomials
Dual Hahn polynomials are related to Hahn polynomials Q by switching the roles of x and n: more precisely
Racah polynomials are a generalization of dual Hahn polynomials