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Let w be a semiclassical weight that is generic in Magnus’s sense, and (pn)n=0∞({p_n})_{n = 0}^\infty the corresponding sequence of orthogonal polynomials. We express the Christoffel–Darboux kernel as a sum of products of Hankel integral operators. For ψ ∈ L∞ (iℝ), let W(ψ) be the Wiener-Hopf operator with symbol ψ. We give sufficient conditions on ψ such that 1/ det W(ψ) W(ψ−1) = det(I − Γϕ1 Γϕ2) where Γϕ1 and Γϕ2 are Hankel operators that are Hilbert–Schmidt. For certain, ψ Barnes’s integral leads to an expansion of this determinant in terms of the generalised hypergeometric 2mF2m-1. These results extend those of Basor and Chen [2], who obtained 4F3 likewise. We include examples where the Wiener–Hopf factors are found explicitly.

This article considers Whittaker's confluent hypergeometric function Wκ,μ where κ is real and μ is real or purely imaginary. Then ϕ(x) = x−μ−1/2Wκ,μ(x) arises as the scattering function of a continuous time linear system with state space L²(1/2, ∞) and input and output spaces C. The Hankel operator Γϕ on L²(0, ∞) is expressed as a matrix with respect to the Laguerre basis and gives the Hankel matrix of moments of a Jacobi weight w0(x) = xb(1 − x)a. The operation of translating ϕ is equivalent to deforming w0 to give wt(x) = e−t/xxb(1 − x)a. The determinant of the Hankel matrix of moments of wε satisfies the σ form of Painlevé's transcendental differential equation PV. It is shown that Γϕ gives rise to the Whittaker kernel from random matrix theory, as studied by Borodin and Olshanski [Commun. Math. Phys. 211, 335-358 (2000)]. Whittaker kernels are closely related to systems of orthogonal polynomials for a Pollaczek-Jacobi type weight lying outside the usual Szegö class. [ABSTRACT FROM AUTHOR]

Abstract: Painlevé''s transcendental differential equation may be expressed as the consistency condition for a pair of linear differential equations with matrix coefficients with rational entries. By a construction due to Tracy and Widom, this linear system is associated with certain kernels which give trace class operators on Hilbert space. This paper expresses such operators in terms of Hankel operators of linear systems which are realised in terms of the Laurent coefficients of the solutions of the differential equations. Let be the orthogonal projection; then the Fredholm determinant defines the τ function, which is here expressed in terms of the solution of a matrix Gelfand–Levitan equation. For suitable values of the parameters, solutions of the hypergeometric equation give a linear system with similar properties. For meromorphic transfer functions that have poles on an arithmetic progression, the corresponding Hankel operator has a simple form with respect to an exponential basis in ; so can be expressed as a series of finite determinants. This applies to elliptic functions of the second kind, such as satisfy Lamé''s equation with . [Copyright &y& Elsevier]

Abstract: In random matrix theory, determinantal random point fields describe the distribution of eigenvalues of self-adjoint matrices from the generalized unitary ensemble. This paper considers symmetric Hamiltonian systems and determines the properties of kernels and associated determinantal random point fields that arise from them; this extends work of Tracy and Widom. The inverse spectral problem for self-adjoint Hankel operators gives sufficient conditions for a self-adjoint operator to be the Hankel operator on from a linear system in continuous time; thus this paper expresses certain kernels as squares of Hankel operators. For suitable linear systems with one-dimensional input and output spaces, there exists a Hankel operator Γ with kernel such that is the generating function of a determinantal random point field on . The inverse scattering transform for the Zakharov–Shabat system involves a Gelfand–Levitan integral equation such that the trace of the diagonal of the solution gives . When is a finite matrix and , there exists a determinantal random point field such that the largest point has a generalised logistic distribution. [Copyright &y& Elsevier]