  The paper considers the distribution of a general linear combination of
central and non-central chi-square random variables by exploring the branch cut
regions that appear in the standard Laplace inversion process. Due to the
original interest from the directional statistics, the focus of this paper is
on the density function of such distributions and not on their cumulative
distribution function. In fact, our results confirm that the latter is a
special case of the former. Our approach provides new insight by generating
alternative characterizations of the probability density function in terms of a
finite number of feasible univariate integrals. In particular, the central
cases seem to allow an interesting representation in terms of the branch cuts,
while general degrees of freedom and non-centrality can be easily adopted using
recursive differentiation. Numerical results confirm that the proposed approach
works well while more transparency and therefore easier control in the accuracy
is ensured.
