Discrete Entropy of Generalized Jacobi Polynomials

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Given a sequence of orthonormal polynomials on R,{pₙ}ₙ≥ ₀, with pₙ of degree n, we define the discrete probability distribution Ψₙ(x) = (Ψₙ,₁(x), Ψₙ,ₙ(x)), with Ψₙ,ⱼ(x) = (∑ⱼ₌₀ⁿ⁻¹ pⱼ²(x))⁻¹ pⱼ₋₁²(x), j=1,, n. In this paper, we study the asymptotic behavior as n→ ∞ of the Shannon entropy S ((Ψₙ(x))= -∑ⱼ₌₁ⁿ Ψₙ,ⱼ(x) (Ψₙ,ⱼ(x)), x∈ (-1,1), when the orthogonality weight is (1-x)α (1+x)β h(x), α, β> -1, and where h is real, analytic, and positive on [-1,1]. We show that the limit ₙ → ∞ (S ((Ψₙ(x))- n) exists for all x∈ (-1,1), but its value depends on the rationality of (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 pₙ, obtained previously in [A.I. Aptekarev, J.S. Dehesa, A. Martinez-Finkelshtein, and R. Yañez, Constr. Approx., 30 (2009), pp. 93-119].