  The statistical distribution of eigenvalues of pairs of coupled random
matrices can be expressed in terms of integral kernels having a generalized
Christoffel--Darboux form constructed from sequences of biorthogonal
polynomials. For measures involving exponentials of a pair of polynomials V_1,
V_2 in two different variables, these kernels may be expressed in terms of
finite dimensional ``windows'' spanned by finite subsequences having length
equal to the degree of one or the other of the polynomials V_1, V_2. The
vectors formed by such subsequences satisfy "dual pairs" of first order systems
of linear differential equations with polynomial coefficients, having rank
equal to one of the degrees of V_1 or V_2 and degree equal to the other. They
also satisfy recursion relations connecting the consecutive windows, and
deformation equations, determining how they change under variations in the
coefficients of the polynomials V_1 and V_2. Viewed as overdetermined systems
of linear difference-differential-deformation equations, these are shown to be
compatible, and hence to admit simultaneous fundamental systems of solutions.
The main result is the demonstration of a spectral duality property; namely,
that the spectral curves defined by the characteristic equations of the pair of
matrices defining the dual differential systems are equal upon interchange of
eigenvalue and polynomial parameters.
