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. References
 
 
 
 

Special hypergeometric functions
Orthogonal polynomials