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

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