Discrete entropies of orthogonal polynomials

Let $p_n$ be the $n$-th orthonormal polynomial on the real line, whose zeros are ${\lambda }_j^{(n)}$, $j=1,\mathrm{.}\mathrm{.}\mathrm{.},n$. Then for each $j=1,\mathrm{.}\mathrm{.}\mathrm{.},n$, $${\vec{\mathrm{\Psi }}}_j^2=({\mathrm{\Psi }}_{1j}^2,\mathrm{.}\mathrm{.}\mathrm{.},{\mathrm{\Psi }}_{nj}^2)$$ with $$ \Psi_{ij}^2= p_{i-1}^2 (\lambda_j^{(n)}) (\sum_{k=0}^{n-1} p_k^2(\lambda_j^{(n)}))^{-1}, \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 \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.