Asymptotics of the information entropy for Jacobi and Laguerre polynomials with varying weights (Q1300151)

The authors investigate the entropy integral \(S_n=-\int_\Delta p^2_{n,n}\ln p^2_{n,n}(x)w_n(x)dx\) asymptotically as \(n\to\infty\). The \(p_{n,n}\)'s are the orthogonal polynomials of degree \(n\) with respect to a Jacobi or Laguerre weight function \(w_n(x)\) with parameters \(\alpha_n=\alpha n+o(n)\), \(\beta_n=\beta n+o(n)\) and \(\alpha,\beta>0\). Their main result shows \(\text{weak}^*\) convergence of the measures \(p^2_{n,n}(x)w_n(x)\) to the Robin distribution of the support of the equilibrium measure in an external field which arises from the limit of the \(w_n^{1/n}(x)\) as \(n\to\infty\). The determination of the position and momentum information entropies of the harmonic oscillator and Coulomb potential are connected with these types of integrals.