Discrete entropies of orthogonal polynomials

Abstract

Let $p_n$ be the $n$-th orthonormal polynomial on the real line, whose zeros are $λ_j^{(n)}$, $j=1, ..., n$. Then for each $j=1, ..., n$, $$ \vec Ψ_j^2 = (Ψ_{1j}^2, ..., Ψ_{nj}^2) $$ with $$ Ψ_{ij}^2= p_{i-1}^2 (λ_j^{(n)}) (\sum_{k=0}^{n-1} p_k^2(λ_j^{(n)}))^{-1}, \quad i=1, >..., n, $$ defines a discrete probability distribution. The Shannon entropy of the sequence $\{p_n\}$ is consequently defined as $$ \mathcal S_{n,j} = -\sum_{i=1}^n Ψ_{ij}^{2} \log (Ψ_{ij}^{2}) . $$ 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.