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 $\ell=0$ Hernquist--Ostriker family, we construct in closed form a set of potential-density pairs whose potentials decay as $r^{-p}$ with a prescribed $p\ge2$, 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 $r\sim n^{-2}$ to $r\sim n^{2}$, 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.