Discrete orthogonal polynomials

In mathematics, a sequence of discrete orthogonal polynomials is a sequence of polynomials that are pairwise orthogonal with respect to a discrete measure.

Examples include the discrete Chebyshev polynomials, Charlier polynomials, Krawtchouk polynomials, Meixner polynomials, dual Hahn polynomials, Hahn polynomials, and Racah polynomials.

If the measure has finite support, then the corresponding sequence of discrete orthogonal polynomials has only a finite number of elements.

The Racah polynomials give an example of this.

with weight function

A family of orthogonal polynomials

is called discrete if they are orthogonal with respect to

is the Kronecker delta.

[1] Any discrete measure is of the form so one can define a weight function by