Dual Hahn polynomials

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

R_n(\lambda(x);\gamma,\delta,N)= {}_3F_2(-n,-x,x+\gamma+\delta+1;\gamma+1,-N;1),\

for 0≤nN 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

 R_n(\lambda(x);\gamma,\delta,N) = Q_x(n;\gamma,\delta,N)

Racah polynomials are a generalization of dual Hahn polynomials

References

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