Discrete Entropy of Generalized Jacobi Polynomials

Abstract

Given a sequence of orthonormal polynomials on $\mathbb{R}$,$\{p_n{\}}_{n\ge 0}$, 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 ${\Psi }_{n,j}(x)=({\sum }_{j=0}^{n-1}{p}_j^2(x){)}^{-1}{p}_{j-1}^2(x)$, $j=1,\dots ,n$. 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 $$\underset{n\to \infty }{\lim }\left(\mathcal{S}(({\Psi }_n(x))-\log n\right)$$ 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. Yañez, Constr. Approx., 30 (2009), pp. 93-119].

12 pages, to appear in Journal of Math. Anal. Appl