The method proposed by Pratt to derive recursion relations for systems of degenerate fermions [S. Pratt, Phys. Rev. Lett. 84, 4255 (2000)] relies on diagrammatic techniques. This efficient formalism assumes no explicit two-body interactions, makes possible the inclusion of conservation laws, and requires low computational time. In this Brief Report, we show that such recursion relations can be obtained from generating functions, without any restriction in relation to the number of conservation laws (e.g., total energy or angular momentum).