A two-parameter family of double-power-law biorthonormal
  potential-density expansions
  Biorthonormal basis function expansions are widely used in galactic dynamics,
both to study problems in galactic stability and to provide numerical
algorithms to evolve collisionless stellar systems. They also provide a compact
and efficient description of the structure of numerical dark matter haloes in
cosmological simulations. We present a two-parameter family of biorthonormal
double-power-law potential-density expansions. Both the potential and density
are given in closed analytic form and may be rapidly computed via recurrence
relations. We show that this family encompasses all the known analytic
biorthonormal expansions: the Zhao expansions (themselves generalizations of
ones found earlier by Hernquist & Ostriker and by Clutton-Brock) and the
recently discovered Lilley, Sanders, Evans & Erkal expansion. Our new
two-parameter family includes expansions based around many familiar spherical
density profiles as zeroth-order models, including the $\gamma$ models and the
Jaffe model. It also contains a basis expansion that reproduces the famous
Navarro-Frenk-White (NFW) profile at zeroth order. The new basis expansions
have been found via a systematic methodology which has wide applications in
finding further examples. In the process, we also uncovered a novel integral
transform solution to Poisson's equation.
