  Efficient expansions of the gravitational field of (dark) haloes have two
main uses in the modelling of galaxies: first, they provide a compact
representation of numerically-constructed (or real) cosmological haloes,
incorporating the effects of triaxiality, lopsidedness or other distortion.
Secondly, they provide the basis functions for self-consistent field expansion
algorithms used in the evolution of $N$-body systems. We present a new family
of biorthogonal potential-density pairs constructed using the Hankel transform
of the Laguerre polynomials. The lowest-order density basis functions are
double-power-law profiles cusped like $\rho \sim r^{-2 + 1/\alpha}$ at small
radii with asymptotic density fall-off like $\rho \sim r^{-3 -1/(2\alpha)}$.
Here, $\alpha$ is a parameter satisfying $\alpha \ge 1/2$. The family therefore
spans the range of inner density cusps found in numerical simulations, but has
much shallower -- and hence more realistic -- outer slopes than the
corresponding members of the only previously-known family deduced by Zhao
(1996) and exemplified by Hernquist & Ostriker (1992). When $\alpha =1$, the
lowest-order density profile has an inner density cusp of $\rho \sim r^{-1}$
and an outer density slope of $\rho \sim r^{-3.5}$, similar to the famous
Navarro, Frenk & White (1997) model. For this reason, we demonstrate that our
new expansion provides a more accurate representation of flattened NFW haloes
than the competing Hernquist-Ostriker expansion. We utilize our new expansion
by analysing a suite of numerically-constructed haloes and providing the
distributions of the expansion coefficients.
