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An integrable system is often formulated as a flat connection, satisfying aLax equation. It is given in terms of compatible systems having a commonsolution called the ``wave function" $\Psi$ living in a Lie group $G$, whichsatisfies some differential equations with rational coefficients. From thiswave function, it is usual to define a sequence of ``correlators" $W_n$, thatplay an important role in many applications in mathematical physics. Here, weshow how to systematically obtain ordinary differential equations (ODE) withpolynomial coefficients for the correlators. An application is random matrixtheory, where the wave functions are the expectation value of thecharacteristic polynomial, they form a family of orthogonal polynomials, andare known to satisfy an integrable system. The correlators are then thecorrelation functions of resolvents or of eigenvalue densities. We give the ODEand recursion on the matrix size that they satisfy.

Recursions and ODEs for correlations in integrable systems and random matrices