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 on a non-uniform lattice ${\displaystyle x(s)=s(s+1)}$ and are defined as

${\displaystyle w_{n}^{(c)}(s,a,b)={\frac {(a-b+1)_{n}(a+c+1)_{n}}{n!}}{}_{3}F_{2}(-n,a-s,a+s+1;a-b+a,a+c+1;1)}$

for ${\displaystyle n=0,1,...,N-1}$ and the parameters ${\displaystyle a,b,c}$ are restricted to ${\displaystyle -{\frac {1}{2}}<a<b,|c|<1+a,b=a+N}$.

Note that ${\displaystyle (u)_{k}}$ is the rising factorial, otherwise known as the Pochhammer symbol, and ${\displaystyle {}_{3}F_{2}(\cdot )}$ is the generalized hypergeometric functions

Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.