Abstract: | Let Q ( x ) = q 2 m x 2 m + q 2 m −1 x 2 m −1+⋯ be a polynomial of degree 2 m with q 2 m > 0 , and let {π n ( x )} n ≥1 be the sequence of monic polynomials orthogonal with respect to the weight w ( x ) = e − Q ( x ) on . Furthermore, let α n and β n denote the Mhaskar–Rakhmanov–Saff (MRS) numbers associated with Q ( x ). By using the Riemann–Hilbert approach, an asymptotic expansion is constructed for π n ( cnz + dn ) , which holds uniformly for all z bounded away from (−∞, −1) , where and . |