Discrete entropy of generalized Jacobi polynomials

Given a sequence of orthonormal polynomials on R, {pn}n≥0, with p_n of degree n, we define the discrete probability distribution Ψn(x)=(Ψn,1(x),...,Ψn,n(x)), with Ψn,j(x)=(∑j=0n-1pj2(x))-1pj-12(x), j=1, ..., n. In this paper, we study the asymptotic behavior as n→∞ of the Shannon entropy S(Ψn(x))=-∑j=1nΨn,j(x)log(Ψn,j(x)), x∈(-1, 1), when the orthogonality weight is (1-x)^α(1+x)^βh(x), α, β>-1, and where h is real, analytic, and positive on [-1, 1]. We show that the limitlimn→∞(S(Ψn(x))-log n) exists for all x∈(-1, 1), but its value depends on the rationality of arccos(x)/π. For the particular case of the Chebyshev polynomials of the first and second kinds, we compare our asymptotic result with the explicit formulas for S(Ψn(ζj(n))), where {ζj(n)} are the zeros of p_n, obtained previously in [2].