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Painleve's transcendental differential equation P_{VI} may be expressed asthe consistency condition for a pair of linear differential equations with 2 by2 matrix coefficients with rational entries. By a construction due to Tracy andWidom, this linear system is associated with certain kernels which give traceclass operators on Hilbert space. This paper expresses such operators in termsof the Hankel operators \Gamma_\phi of linear systems which are realised interms of the Laurent coefficients of the solutions of the differentialequations. Let P_{(t infty)}:L^2(0, \infty)\to L^2(t, \infty) be the orthogonalprojection. For such, the Fredholm determinant \tau (t)=det (I-P_{(t,\infty)}\Gamma_\phi) defines the tau function, which is here expressed in termsof the solutions of a matrix Gelfand--Levitan equation. For suitable paramters,solutions of the hypergeometric equation give a linear system with similarproperties. For meromorphic transfer functions \hat\phi that have poles on anarithmetic progression, the corresponding Hankel operator has a simple formwith respect to an exponential basis in L^2(0, \infty); so \tau (t) can beexpressed in terms of finite determinants. This applies to elliptic functionsof the second kind, such as satisfy Lame's equation with \ell=1.

On linear systems and tau functions associated with Lame's equation and Painleve's equation VI