Asymptotics of recurrence coefficients for the Laguerre weight with a singularity at the edge

Abstract: In this paper, We study the asymptotics of the leading coefficients and the recurrence coefficients for the orthogonal polynomials with repect to the Laguerre weight with singularity of root type and jump type at the soft edge via the Deift-Zhou steepest descent method. The asymptotic formulas of the leading coefficients and the recurrence coefficients for large n are described in terms of a class of analytic solutions to the the {\sigma}-form of the Painlev\'e II equation and the Painlev\'e XXXIV equation.