  Painleve's transcendental differential equation P_{VI} may be expressed as
the consistency condition for a pair of linear differential equations with 2 by
2 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 the Hankel operators \Gamma_\phi of linear systems which are realised in
terms of the Laurent coefficients of the solutions of the differential
equations. Let P_{(t infty)}:L^2(0, \infty)\to L^2(t, \infty) be the orthogonal
projection. For such, the Fredholm determinant \tau (t)=det (I-P_{(t,
\infty)}\Gamma_\phi) defines the tau function, which is here expressed in terms
of the solutions of a matrix Gelfand--Levitan equation. For suitable paramters,
solutions of the hypergeometric equation give a linear system with similar
properties. For meromorphic transfer functions \hat\phi that have poles on an
arithmetic progression, the corresponding Hankel operator has a simple form
with respect to an exponential basis in L^2(0, \infty); so \tau (t) can be
expressed in terms of finite determinants. This applies to elliptic functions
of the second kind, such as satisfy Lame's equation with \ell=1.
