We investigate the asymptotic behavior of the smallest eigenvalue, À, of two cat-egories of (N+1)×(N+1) Hankel (or moments) matrices, denoted by Hₙ=(μₘ₊ₙ)ᴺₘ,ₙ₌₀. Firstly, we dealt with the Hankel matrix generated by the Jacobi weight (also called Beta density) xᵃ(1 -x)ᵝ, x ∈ [0, 1], α,β > -1. By applying the arguments of Szegö, Widom and Wilf, we establish the asymptotic formula for the orthonormal polynomials Pₙ(z),z ∈ ℂ\[0, 1], associated with the Jacobi weight, which are required in the determination of λₙ. Based on this formula, we produce the expressions for λₙ, for large N. The second Hankel matrix we studied is with respect to the deformed Laguerre weight (Gamma density) xᵃe⁻ˣᵇ, x∈[0, ∞), α >-1, β > ½. Based on the research by Szegö, Chen, etc., we obtain an asymptotic expression of the orthonormal polyno-mials Pₙ(z)as N⭢∞, associated with this Gamma density. Using this, we obtain the specific asymptotic formulas of the smallest eigenvalue in this paper. Applying the parallel algorithm discovered by Emmart, Chen and Weems, we get a variety of numerical results of ÀN corresponding to our theoretical calculations. It shows that the theoretical results are in close proximity to the numerical results for sufficiently large N. Finally, we study the recurrence coefficients of the monic polynomials Pₙ(z) or-thogonal with respect to the deformed Freud weight wₐ(x; s, N) = |x|ᵃe⁻ᴺ[x²+s(x⁴-x²)], エ € R, with parameters a >-l,N>0,s € [0, 1]. We show that the recurrence coeffi-cients βₙ(s) satisfy the first discrete Painlevé equation (denoted by dPi), a differential-difference equation and a second order nolinear ordinary differential equation (ODE)in s. Here n is the order of the Hankel matrix generated by wₐ (z, s, N). We describe the asymptotic behavior of the recurrence coefficients in three situations, (i)s ⭢ 0,n, N finite, (ii)n → ∞, N finite, (iii) n, N ⭢ ∞, such that the radio r := n/N is bound-ed away from 0 and closed to 1. We also investigate the existence and uniqueness for the positive solutions of the dPi. Further more, we derive, using the ladder operator approach, a second order linear ODE satisfied by the polynomials Pₙ(z). It is found as n⭢ ∞, the linear ODE turns to be a biconfluent Heun equation. This paper concludes with the study of the Hankel determinant, D,(s), associated with wₐ(x; s, N) when n tends to infinity.