Entropy of Orthogonal Polynomials with Freud Weights and Information Entropies of the Harmonic Oscillator Potential

The information entropy of the harmonic oscillator potential V (x) = 1 2 x 2 in both position and momentum spaces can be expressed in terms of the so-called \entropy of Hermite polynomials", i.e. the quantity S n (H) := R +1 ?1 H 2 n (x) log H 2 n (x)e ?x 2 dx. These polynomials are instances of the poly-nomials orthogonal with respect to the Freud weights w(x) = exp(?jxj m), m > 0. Here, rstly the leading term of the entropy of Freud polynomials is found by use of the theory of strong asymptotics of orthogonal polynomials.