We study the recurrence coefficients of the monic polynomials orthogonal with respect to the deformed (also called semi-classical) Freud weight wα(x; s,N) = |x|αe−N[x2+s(x4−x2)],x ∈ ℝ, with parameters . We show that the recurrence coefficients satisfy the first discrete Painlevé equation (denoted by d), a differential–difference equation and a second-order nonlinear ordinary differential equation (ODE) in . Here is the order of the Hankel matrix generated by . We describe the asymptotic behavior of the recurrence coefficients in three situations: (i) , finite, (ii) , finite, (iii) , such that the radio is bounded away from and closed to . We also investigate the existence and uniqueness for the positive solutions of the d. Furthermore, we derive, using the ladder operator approach, a second-order linear ODE satisfied by the polynomials . It is found as , the linear ODE turns to be a biconfluent Heun equation. This paper concludes with the study of the Hankel determinant, , associated with when tends to infinity.