Discrete Entropy of Generalized Jacobi Polynomials

TL;DR

This paper investigates the asymptotic behavior of the Shannon entropy associated with discrete probability distributions derived from generalized Jacobi polynomials, revealing dependence on the rationality of a specific angle related to the polynomial's evaluation point.

Contribution

It provides the first asymptotic analysis of the discrete entropy for generalized Jacobi polynomials, including explicit limits depending on the rationality of arccos(x)/pi, and compares with known explicit formulas for Chebyshev cases.

Findings

The entropy limit exists for all x in (-1,1).

The limit depends on whether arccos(x)/pi is rational or irrational.

Explicit formulas are confirmed for Chebyshev 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].