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Under the separability assumption on the augmented density, a distributionfunction can be always constructed for a spherical population with thespecified density and anisotropy profile. Then, a question arises, under whatconditions the distribution constructed as such is non-negative everywhere inthe entire accessible subvolume of the phase-space. We rediscover necessaryconditions on the augmented density expressed with fractional calculus. Thecondition on the radius part R(r^2) -- whose logarithmic derivative is theanisotropy parameter -- is equivalent to R(1/w)/w being a completely monotonicfunction whereas the condition on the potential part is stated as itsderivative up to the order not greater than 3/2-b being non-negative (where bis the central limiting value for the anisotropy parameter). We also derive theset of sufficient conditions on the separable augmented density for thenon-negativity of the distribution, which generalizes the condition derived forthe generalized Cuddeford system by Ciotti & Morganti to arbitrary separablesystems. This is applied for the case when the anisotropy is parameterized by amonotonic function of the radius of Baes & Van Hese. The resulting criteria arefound based on the complete monotonicity of generalized Mittag-Lefflerfunctions.

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