A radial basis of massless potential-density pairs

In the matrix method for linear perturbations of spherical stellar systems, the perturbed potential and density are expanded over a set of potential-density pairs. For radial perturbations, mass conservation makes the coefficient of 1/r in the perturbed potential vanish, so that the potential decays faster than 1/r. Starting from the =0 Hernquist--Ostriker family, we construct in closed form a set of potential-density pairs whose potentials decay as r-p with a prescribed p 2, the tail containing all subsequent integer powers. For p=2 each pair is individually massless, and the potential of the n-th pair is expressed through a single Jacobi polynomial. We derive the combination weights, the leading tail coefficients and the Gram matrix analytically; the Gram matrix is banded with half-width p-1 (tridiagonal at p=2). The n-th potential element has exactly n-1 nodes, spread from r about n-2 to r about n squared, so that the set resolves both the centre and the far periphery. As a test, the expansion of the dilation-mode potential of the isochrone model converges exponentially and carries no parasitic mass at any truncation, in contrast to the standard set.