Discrete entropies of orthogonal polynomials

Abstract

Let $p_n$ be the $n$-th orthonormal polynomial on the real line, whose zeros are ${\lambda }_j^{(n)}$, $j=1,...,n$. Then for each $j=1,...,n$, $${\vec{\Psi }}_j^2=({\Psi }_{1j}^2,...,{\Psi }_{nj}^2)$$ with $${\Psi }_{ij}^2=p_{i-1}^2({\lambda }_j^{(n)})(\sum _{k=0}^{n-1}{p}_k^2({\lambda }_j^{(n)}){)}^{-1},i=1,>...,n,$$ defines a discrete probability distribution. The Shannon entropy of the sequence $\{p_n\}$ is consequently defined as $${\mathcal{S}}_{n,j}=-\sum _{i=1}^n{\Psi }_{ij}^2\log ({\Psi }_{ij}^2).$$ In the case of Chebyshev polynomials of the first and second kinds an explicit and closed formula for ${\mathcal{S}}_{n,j}$ is obtained, revealing interesting connections with the number theory. Besides, several results of numerical computations exemplifying the behavior of ${\mathcal{S}}_{n,j}$ for other families are also presented.

26 pages, 6 figures