# Orthogonal polynomials with respect to discrete probability distributions

The polynomials from the Askey scheme are orthogonal with respect to a standard probability distribution. For example, Hermite polynomials (in a suitable form) are orthogonal with respect to a standard Gaussian distribution, Legendre polynomials are orthogonal with respect to a Uniform$$(-1,1)$$ distribution, etc. I do not understand how there can exist a sequence of orthogonal polynomials with respect to a discrete probability distribution with finitely many point masses. For example, let $$Z\sim\text{Binomial}(3,0.2)$$. Then $$Z$$ can take four values, so there cannot exist more than five polynomials orthogonal with respect to $$Z$$, right?

If one has five orthogonal polynomials $$\phi_0(Z),\ldots,\phi_4(Z)$$, each $$\phi_i(Z)$$ of degree $$i$$, then the linear combination $$a_0\phi_0(Z)+\ldots+a_4\phi_4(Z)=0$$ may hold with some $$a_j\neq0$$, since a non-zero fourth-degree polynomial can vanish at four values. So $$\phi_0(Z),\ldots,\phi_4(Z)$$ are not orthogonal.

• what is your question exactly ? i am not sure if you are making a statement or asking a question, thanks! – Ezy Nov 14 '18 at 3:50
• It is not super nice to attract people with a bounty offer and let it expire without awarding it – Ezy Nov 14 '18 at 17:21

## 1 Answer

discrete ensembles with finite support have a finite rather than infinite family of orthogonal polynomials indeed

see here for reference on discrete orthogonal polynomials

https://en.wikipedia.org/wiki/Discrete_orthogonal_polynomials

you can also check this paper that discusses finite set of orthogonal polynomials and the Askey scheme in particular (section 6), hope this helps!

Orthogonal polynomials, a short introduction (Koornwinder, 2013)

Another simple example is the Lagrange interpolation polynomials which form a finite set of orthogonal polynomials with respect to the canonical discrete measure formed by a discrete set.