The Altenberger-Dahler positive-function renormalization group (PFRG) method is shown to yield the universal scaling equation D-s = D-o exp(-alpha c(v)) of the hydrodynamic scaling model for polymer self-diffusion. Here D-o is the bare polymer self-diffusion coefficient at some low concentration, c is the (potentially high) polymer concentration, and v and alpha are a scaling coefficient and scaling prefactor, respectively. To integrate the Lie equations of motion of the PFRG and obtain the universal scaling equation, the Kirkwood-Risemann model for polymer hydrodynamics is extended analytically to determine leading terms of the chain-chain and (for the first time) chain-chain-chain translation-translation hydrodynamic interaction tensors b(ij)((2)), b(ijl)((3)), T-ij((2)), T-ijl((3)), and T-ijkl((4)), as well as many of their translational-rotational and rotational-rotational analogues.