We are concerned with the probability that all the eigenvalues of a unitary ensemble with the weight function , are greater than s. This probability is expressed as the quotient of D-n(s,t) and its value at s = 0, where D-n(s,t) denotes the determinant of the n dimensional Hankel matrices generated by the moments of w(x;t) on x is an element of [s, infinity). In this paper we focus specifically on the Hankel determinant D-n(s,t) and its properties. Based on the ladder operators adapted to the monic polynomials orthogonal with respect to w(x;t), and from the associated supplementary conditions and a sum-rule, we show that the log-derivative of the Hankel determinant, viewed as a function of s and t, satisfies a second order sixth degree partial differential equation, where n appears as a parameter. In order to go to the thermodynamic limit, of infinitely large matrices, we envisage a scenario where n -> infinity, s -> 0, and t -> 0 such that S := 4ns and T := (2n + 1 + alpha)t are finite. After such a double scaling, the large finite n equation reduces to a second order second degree equation, in the variables S and T, from which we derive the asymptotic expansion of the scaled Hankel determinant in three cases of S and T : S -> infinity with T fixed, S -> 0 with T > 0 fixed, and T -> infinity with S > 0 fixed. The constant term in the asymptotic expansion is shown to satisfy a difference equation and one of its solutions is the Tracy-Widom constant.