We are concerned with the probability that all the eigenvalues of a unitary ensemble with the weight function w(x;t) = x α e −x− t x , x ∈ [0, ∞), α > −1, t ≥ 0, 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 ∈ [s, ∞). 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 → ∞, s → 0, and t → 0 such that S := 4ns and T := (2n + 1 + α)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 → ∞ with T fixed, S → 0 with T > 0 fixed, and T → ∞ 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.

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Akhiezer Polynomials orthogonal on several intervals are used to define a generalization of the beta integral where the integral is over two disjoint intervals of the real line, [−1, −β] ∪ [β, 1]. An explicit evaluation of the integral is given in the limiting case as β → 1.

We give an explicit formula for the expansion coefficients of a generalized beta integral on the set [−1, −b] ∪ [b, 1] b ∈ (0, 1), in a power series in the parameter b, thus defining a generalized beta function of two complex variables.