The use of primitive Hermite Gaussian auxiliary functions for the variational fitting of the Coulomb potential is discussed. Integral recurrence relations for three-center electron repulsion integrals and integral sums with these auxiliary functions are derived. An asymptotic expansion of three-center electron repulsion integrals with primitive Hermite Gaussian auxiliary functions is presented. The recurrence relations for the asymptotic expanded integrals are given. Integral pathway diagrams and density matrix assignment diagrams are presented for all target integrals. The calculations of the three-center electron repulsion integrals of alkenes indicate that the new algorithms represent a substantial performance improvement. (C) 2003 American Institute of Physics.