Let $\mathbb{X}_{p,n}$ be a $(p \times n)$ random matrix whose entries are iid complex standard normal random variables. The hermitian random matrix $\mathbb{S}_{p,n} = \frac{1}{n} \mathbb{X}_{p,n} \mathbb{X}_{p,n}^*$ is said to have isotropic complex Wishart distribution. Is the exact formula for the trace moments $\mathbb{E}\big[ \operatorname{tr}\big( \mathbb{S}_{p,n}^{l} \big) \big]$ known?
With Harer-Zagier recursion the trace moments of GUE matrices $\mathbb{A}_n$ were shown to be \begin{align*} & \mathbb{E}\big[ \operatorname{tr}\big( \mathbb{A}_{n}^{2l} \big) \big] = \frac{(2l)!}{n^l \, l!} \sum\limits_{\substack{r=1}}^{\max(n,l+1)} \frac{1}{2^{l-r+1}} {l \choose r-1} {n \choose r} \end{align*} with all odd moments being zero. Is there a similarly elegant formula (preferrably explicit and not relying on sums over partitions) for the complex Wishart case?
The two ensembles seem very related and a lot is known about Wishart matrices, so I would have thought such a formula to exist, but can't find it in the huge amount of works dealing with Wishart matrices.
Any tips or references are much apprechiated!
Edit: Using some older generalizations of Harer-Zagier one can in a few pages of calculations arrive at the Formula \begin{align*} & \mathbb{E}\Big[ \operatorname{tr}\big( \mathbb{S}_{p,n}^{l} \big) \Big]\\ & = \frac{l!}{n^l} \sum\limits_{r=1}^{(p+n) \land (l+1)} \sum\limits_{b=1 \lor (r-n)}^{p \land (r-1)} {p \choose b} {n \choose r-b} {l-1 \choose b-1, r-b-1, l-r+1} \ . \end{align*} Simulations seem to agree with this formula. This must be already known, I just couldn't find it in the literature.
Second edit: I found a source: Corollary 1.9 in "Moments of Normally Distributed Random Matrices Given by Generating Series for Connection Coefficients — Explicit Bijective Computation" by Ekaterina Vassilieva.
