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.

  • $\begingroup$ 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

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

see here for reference on 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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