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

Note that

is the rising factorial, otherwise known as the Pochhammer symbol, and

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

The dual Hahn polynomials have the orthogonality condition for

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

The Hahn polynomials,

; α , β )

, is defined on the uniform lattice

a = ( α + β )

, c = ( β − α )

α = β = 0

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.