New biorthogonal potential–density basis functions

We use the weighted integral form of spherical Bessel functions and introduce a new analytical set of complete and biorthogonal potential–density basis functions. The potential and density functions of the new set have finite central values and they fall off, respectively, similar to   r ^{?(1+ l )}  and   r ^{?(4+ l )}  at large radii, where l is the latitudinal quantum number of spherical harmonics. The lowest order term associated with   l = 0  is the perfect sphere of de Zeeuw. Our basis functions are intrinsically suitable for the modelling of three-dimensional, soft-centred stellar systems and they complement the basis sets of Clutton-Brock, Hernquist & Ostriker and Zhao. We test the performance of our functions by expanding the density and potential profiles of some spherical and oblate galaxy models.