  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.
