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We prove that the correlations functions, generated by the determinantal process of the Christoffel-Darboux kernel of an arbitrary order 2 ODE, do satisfy loop equations.
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Y. Fyodorov, E. Strahov, “An exact formula for general spectral correlation function of random Hermitian matrices”, J.Phys.A36:3203-3214,2003, math-ph/0204051
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