Discrete entropy of generalized Jacobi polynomials

Given a sequence of orthonormal polynomials on RR, {pn}n≥0{pn}n≥0, with pnpn of degree n  , we define the discrete probability distribution Ψn(x)=(Ψn,1(x),…,Ψn,n(x))Ψn(x)=(Ψn,1(x),…,Ψn,n(x)), with Ψn,j(x)=(∑j=0n−1pj2(x))−1pj−12(x), j=1,…,nj=1,…,n. In this paper, we study the asymptotic behavior as n→∞n→∞ of the Shannon entropy S(Ψn(x))=−∑j=1nΨn,j(x)log⁡(Ψn,j(x)), x∈(−1,1)x∈(−1,1), when the orthogonality weight is (1−x)α(1+x)βh(x), α,β>−1α,β>−1, and where h   is real, analytic, and positive on [−1,1][−1,1]. We show that the limitlimn→∞⁡(S(Ψn(x))−log⁡n) exists for all x∈(−1,1)x∈(−1,1), but its value depends on the rationality of arccos⁡(x)/πarccos⁡(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 pnpn, obtained previously in [2].