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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.

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

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