A generalization of the Hiller-Sucher-Feinberg (HSF) and Rassolov-Chipman identities for the electron density is presented. A companion expression for the electrostatic potential is derived. Properties of the electron density furnished by the generalized HSF identity are discussed. In particular, the behavior in the vicinity of an arbitrary potential singularity and the long-range asymptotics are analyzed in detail. A simple numerical example is provided to illustrate the new theoretical results.