We review some recursive Fermi operator expansion techniques for the calculation of the density matrix and its response to perturbations in tight-binding, Hartree-Fock, or density functional theory, at zero or finite electronic temperatures. Thanks to the recursive formulation, the expansion order increases exponentially with the number of iterations and the computational cost scales only linearly with the system size for sufficiently large sparse matrix representations. The methods are illustrated using simple models that are suitable for small numerical experiments.