In this paper, we study the recurrence coefficients of a deformed or semi-classical Laguerre polynomials orthogonal with respect to the weight w(x,s) = w(x; := xe-N(x+s(x2-x)), 0 ≤ x < ∞. Here >-1, 0 ≤ s ≤ 1 and N > 0. We will describe this problem in terms of the ratio r:= n N, where ultimately r is bounded away from 0, and close to 1. From the ladder operator approach, and the associated compatibility conditions, the recurrence coefficients satisfy a second order ordinary differential equation (ODE) when viewed as functions of the parameter s in the weight. Then we work out the large degree asymptotics of their recurrence coefficients. We also discuss the associated Hankel determinant. We show that the logarithmic derivative of Dn(s) can be expressed in terms of the recurrence coefficients, and obtained the large degree asymptotics of ln[Dn(1)]. Based on this result, we compute the probability that an n × n (or rN × rN) random matrix from a generalized Gaussian Unitary Ensemble (gGUE) is positive definite.