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.

Definition

Consider a discrete measure $\mu$ on some set $S=\$ with weight function $\omega(x)$.

A family of orthogonal polynomials $\$ is called discrete, if they are orthogonal with respect to $\omega$ (resp. $\mu$), i.e.
:$\sum\limits_ p_n(x)p_m(x)\omega(x)=\kappa_n\delta_,$
where $\delta_$ is the Kronecker delta.

Remark

Any discrete measure is of the form
: $\mu = \sum_ a_i \delta_$,
so one can define a weight function by $\omega(s_i) = a_i$.

Listeratur

*{{Citation , last1=Baik , first1=Jinho , last2=Kriecherbauer , first2=T. , last3=McLaughlin , first3=K. T.-R. , last4=Miller , first4=P. D. , title=Discrete orthogonal polynomials. Asymptotics and applications , url=https://books.google.com/books?id=S-GIGgx05kgC , publisher= Princeton University Press , series=Annals of Mathematics Studies , isbn=978-0-691-12734-7 , mr=2283089 , year=2007 , volume=164

References

Orthogonal polynomials