Discrete Entropy of Generalized Jacobi Polynomials

Abstract

Given a sequence of orthonormal polynomials on $\Bbb R$,$\{p_n\}_{n\geq 0}$, with $p_n$ of degree $n$, we define the discrete probability distribution $\Psi_n(x) = \left(\Psi_{n,1}(x), \dots \Psi_{n,n}(x) \right) $, with $\Psi_{n,j}(x) = \big(\sum_{j=0}^{n-1} p_j^2(x)\big)^{-1} p_{j-1}^2(x)$, $j=1, \dots, n$. In this paper, we study the asymptotic behavior as $n\to \infty$ of the Shannon entropy $\mathcal S ((\Psi_n(x))= -\sum_{j=1}^n \Psi_{n,j}(x) \log (\Psi_{n,j}(x))$, $x\in (-1,1)$, when the orthogonality weight is $ (1-x)^{\alpha}\, (1+x)^{\beta}\, h(x) $, $\alpha, \beta > -1$, and where $h$ is real, analytic, and positive on $[-1,1]$. We show that the limit $$ \lim_{n \to \infty} \left(\mathcal{S} ((\Psi_n(x))- \log n\right) $$ exists for all $x\in (-1,1)$, but its value depends on the rationality of $\arccos(x)/\pi$. For the particular case of the Chebyshev polynomials of the first and second kinds, we compare our asymptotic result with the explicit formulas for $\mathcal{S} (\Psi_n(\zeta_j^{(n)}))$, where $\{\zeta_j^{(n)}\}$ are the zeros of $p_n$, obtained previously in [A.I. Aptekarev, J.S. Dehesa, A. Martinez-Finkelshtein, and R. Ya\~nez, Constr. Approx., 30 (2009), pp. 93-119].