Consider a discrete measure ${\displaystyle \mu }$ on some set ${\displaystyle S=\{s_{0},s_{1},\dots \}}$ with weight function ${\displaystyle \omega (x)}$.

A family of orthogonal polynomials ${\displaystyle \{p_{n}(x)\}}$ is called discrete if they are orthogonal with respect to ${\displaystyle \omega }$ (resp. ${\displaystyle \mu }$), i.e.,

${\displaystyle \sum \limits _{x\in S}p_{n}(x)p_{m}(x)\omega (x)=\kappa _{n}\delta _{n,m},}$

where ${\displaystyle \delta _{n,m}}$ is the Kronecker delta.^[1]

Any discrete measure is of the form

${\displaystyle \mu =\sum _{i}a_{i}\delta _{s_{i}}}$,

so one can define a weight function by ${\displaystyle \omega (s_{i})=a_{i}}$.

• Baik, Jinho; Kriecherbauer, T.; McLaughlin, K. T.-R.; Miller, P. D. (2007), Discrete orthogonal polynomials. Asymptotics and applications, Annals of Mathematics Studies, vol. 164, Princeton University Press, ISBN 978-0-691-12734-7, MR 2283089