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

We consider orthogonal polynomials {pn,N(x)}n=0 on the real line with respect to a weight w(x) = e (x) and in particular the asymptotic behaviour of the coefficients an,N and bn,N in the three term recurrence xπn,N (x) = πn+1,N (x)+bn,Nπn,N (x)+an,Nπn−1,N (x). For one-cut regular V we show, using the Deift-Zhou method of steepest descent for Riemann-Hilbert problems, that the diagonal recurrence coefficients an,n and bn,n have asymptotic expansions as n → ∞ in powers of 1/n and powers of 1/n, respectively.