Asymptotics for entropy integrals of orthogonal polynomials on the unit circle

The asymptotics for entropy integrals of orthogonal polynomials on real line were investigated recently to determine the Boltzmann-Shannon information entropy of quantum-mechanical systems in central potentials. The entropy integrals for classical orthogonal polynomials are defined as E_n = -∫ p_n²(x) log p_n²(x)w(x)dx. Here, we consider the entropy-type of integrals of orthogonal polynomials on the unit circle E_n(|z|) = -int;₀^2π |φ_n² (e^iθ)|log|φ_n² (e^iθ)dσ(θ), where (φ_n(z)) are orthonormal polynomials with respect to the measure dσ(θ) on the unit circle |z| = 1. An asymptotic formula for E_n(|z|) is obtained in terms of dσ(θ). We also determine the asymptotics of entropy integrals for orthogonal polynomials on a rectifiable Jordan curve Γ. Furthermore we give explicit formulas of entropy integrals for Fibonacci and Lucas polynomials on [-2i, 2i].