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 and are defined as
for and the parameters are restricted to.
Note that is the rising factorial, otherwise known as the Pochhammer symbol, and is the generalized hypergeometric functions
give a detailed list of their properties.

Orthogonality

The dual Hahn polynomials have the orthogonality condition
for. Where,
and

Numerical instability

As the value of increases, the values that the discrete polynomials obtain also increases. As a result, to obtain numerical stability in calculating the polynomials you would use the renormalized dual Hahn polynomial as defined as
for.
Then the orthogonality condition becomes
for

Relation to other polynomials

The Hahn polynomials,, is defined on the uniform lattice, and the parameters are defined as. Then setting the Hahn polynomials become the Chebyshev polynomials. Note that the dual Hahn polynomials have a q-analog with an extra parameter q known as the dual q-Hahn polynomials.
Racah polynomials are a generalization of dual Hahn polynomials.