The non-central chi-square distribution with $\nu$ degrees of freedom has a density which can be expressed as
$$
f(x) = \sum_{i=0}^{\infty} c_i f_{\nu + 2i}(x),
$$
where $c_i$ is a function of the non-centrality parameter and $i$, and where $f_{j}(x)$ is the density of a chi-square distribution with $j$ degrees of freedom. (*The non-central chi-squared distribution is a Poisson weighted mixture of central chi-squared distributions*).

My question is whether this property extends to the multivariate case, the non-central Wishart distribution: can we express the non-central Wishart as a weighted mixture of central Wishart distributions?

(My guess is not, except in the scalar case; if it does hold, however, and the $c_i$ are easy to compute, I would like to use this property to compute the moments of the non-central Wishart.)