Asymptotics of the information entropy for Jacobi and Laguerre polynomials with varying weights

We study the asymptotic behavior as n --> infinity of the entropy integral S-n = - integral p(n, n)(2)(x) ln p(n, n)(2)(x) w(n)(x) dx, where p(n, n) is the nth degree polynomial orthonormal with respect to a Jacobi or Laguerre weight function w(n)(x) whose parameters grow with n. For this purpose we use the weak-* convergence of the measures p(n, n)(2)(x)w(n)(x) to the Robin distribution of the support of the equilibrium measure in an external field, arising from the limit of the nth root of the sequence of weights. (C) 1999 Academic Press.