1201.6113 (J. An)

Fractional calculus, completely monotonic functions, a generalized
Mittag-Leffler function and phase-space consistency of separable augmented
densities    [PDF]

J. An

Under the separability assumption on the augmented density, a distribution
function can be always constructed for a spherical population with the
specified density and anisotropy profile. Then, a question arises, under what
conditions the distribution constructed as such is non-negative everywhere in
the entire accessible subvolume of the phase-space. We rediscover necessary
conditions on the augmented density expressed with fractional calculus. The
condition on the radius part R(r^2) -- whose logarithmic derivative is the
anisotropy parameter -- is equivalent to R(1/w)/w being a completely monotonic
function whereas the condition on the potential part is stated as its
derivative up to the order not greater than 3/2-b being non-negative (where b
is the central limiting value for the anisotropy parameter). We also derive the
set of sufficient conditions on the separable augmented density for the
non-negativity of the distribution, which generalizes the condition derived for
the generalized Cuddeford system by Ciotti & Morganti to arbitrary separable
systems. This is applied for the case when the anisotropy is parameterized by a
monotonic function of the radius of Baes & Van Hese. The resulting criteria are
found based on the complete monotonicity of generalized Mittag-Leffler
functions.

View original: http://arxiv.org/abs/1201.6113