Asymptotics for entropy integrals of orthogonal polynomials on the unit circle

The asymptotics for entropy integrals of orthogonal polynomials on real lin e were investigated recently to determine the Boltzmann-Shannon information entropy of quantum-mechanical systems in central potentials. The entropy i ntegrals for classical orthogonal polynomials are defined as E-n = -integralp(n)(2)(x) log p(n)(2)(x)w(x)dx. Here, we consider the entropy-type of integrals of orthogonal polynomials o n the unit circle E-n(/z/) = integral (2 pi)(0)/phi (2)(n)(e(i theta))/ log /theta (2)(n)(e(i theta))d sigma(theta) where {phi (n)(z)} are orthonormal polynomials with respect to the measure d sigma(theta) on the unit circle /z/ = 1. An asymptotic formula for E-n(/z /) is obtained in terms of d sigma(theta). We also determine the asymptotic s of entropy integrals for orthogonal polynomials on a rectifiable Jordan c urve Gamma. Furthermore we give explicit formulas of entropy integrals for Fibonacci and Lucas polynomials on [-2i, 2i].