Global asymptotics of orthogonal polynomials associated with |x|2αe−Q(x)

In this paper, we consider the asymptotics of polynomials orthogonal with respect to the weight function w(x)=|x|2αe−Q(x),α>−12, where Q(x)=∑k=02mqkxk,q2m>0,m>0 is a polynomial of degree 2m2m. Globally uniform asymptotic expansions are obtained for zz in four regions. These regions together cover the whole complex zz-plane. Due to the singularity of |x|2α|x|2α, the expansion in the region containing the origin involves Bessel functions. We also study the asymptotic behavior of the leading coefficients and the recurrence coefficients of these polynomials. Our approach is based on a modified version of the steepest descent method for Riemann–Hilbert problems introduced by Deift and Zhou [P. Deift, X. Zhou, A steepest descent method for oscillatory Riemann–Hilbert problems, Asymptotics for the mKdV equation, Ann. of Math. 137 (1993) 295–368].