  A sum rule is an identity connecting the entropy of a measure with
coefficients involved in the construction of its orthogonal polynomials (Jacobi
coefficients). Our paper is an extension of Gamboa, Nagel and Rouault (2016),
where we have showed sum rules by using only probabilistic tools (namely the
large deviations theory). Here, we prove large deviation principles for the
weighted spectral measure of unitarily invariant random matrices in two general
situations: firstly, when the equilibrium measure is not necessarily supported
by a single interval and secondly, when the potential is a nonnegative
polynomial. The rate functions can be expressed as functions of the Jacobi
coefficients. These new large deviation results lead to original sum rules both
for the one and the multi-cut regime and also answer a conjecture stated in
Gamboa, Nagel and Rouault (2016) concerning general sum rules.
