  We consider ensembles of Gaussian (Hermite) and Wishart (Laguerre) $N\times
N$ hermitian matrices. We study the effect of finite rank perturbations of
these ensembles by a source term. The rank $r$ of the perturbation corresponds
to the number of non-null eigenvalues of the source matrix. In the perturbed
ensembles, the correlation functions can be written in terms of kernels. We
show that for all $N$, the difference between the perturbed and the unperturbed
kernels is a degenerate kernel of size $r$ which depends on multiple Hermite or
Laguerre functions. We also compute asymptotic formulas for the multiple
Laguerre functions kernels in terms multiple Bessel (resp. Airy) functions.
This leads to the large $N$ limiting kernels at the hard (resp. soft) edge of
the spectrum of the perturbed Laguerre ensemble. Similar results are obtained
in the Hermite case.
