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 formula_1 and are defined as
for formula_3 and the parameters formula_4 are restricted to formula_5 is the falling and rising factorials, otherwise known as the Pochhammer symbol, and formula_6 is the generalized hypergeometric functions
 give a detailed list of their properties.
Orthogonality.
The Dual Hahn polynomials have the orthogonality condition
for formula_8. Where formula_9,
and
Numerical Instability.
As the value of formula_12 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 renormalised Dual Hahn Polynomial as defined as
for formula_3.
Then the orthogonality condition becomes
for formula_8
Relation to other polynomials.
The Hahn polynomials, formula_17, is defined on the uniform lattice formula_18, and the parameters formula_4 are defined as formula_20. Then setting formula_21 the Hahn polynomials become the Tchebichef polynomials. Note that the Dual Hahn polynomials have a "q"-analog with an extra parameter "q" known as the Dual Hahn Q-polynomials 
Racah polynomials are a generalization of dual Hahn polynomials