In this talk I will explain a set of combinatorial identities combinatorial identities for the set of partitions. This identities are motivated by the study for certain limit theorems related to finite free convolutions trough finite free cumulants. In particular, this allows to give a combinatorial proof of the recent result of Kabluchko describing the asymptotic behaviour of the roots of Unitary Hermite and Laguerre Polynomials. A connection with classical cumulants of polynomials of a Poisson variable will also be explained.