The asymptotic behaviour of recurrence coefficients for orthogonal polynomials with varying exponential weights (Q1035632)

Let \(\pi_{n,N}(x)\) be the \(n\)-th degree monic orthogonal polynomial with respect to the weight \(e^{-NV(x)}\). Under some assumptions on \(V(x)\) the authors study the asymptotic behaviour of the coefficients \(a_{n,N}\) and \(b_{n,N}\) in the three-term recurrence relation \[ \chi \pi_{n,N}(x)= \pi_{n+1,N}(x)+b_{n,N}\pi_{n,N}(x)+a_{n,N}\pi_{n-1,N}(x) \] as \(n\to\infty.\) The method is based on the so called steepest descent for \(2\times 2\)- Riemann-Hilbert boundary problem associated with the orthogonal polynomials under consideration.