Spectral moments of the real Ginibre ensemble (Q6596373)

An \(N\times N\) real Ginibre matrix has both real eigenvalues \((x_i)\) and complex eigenvalues \((z_i)\). The distribution and number of these eigenvalues was studied, for instance in [\textit{A. Edelman}, J. Multivariate Anal. 60, No. 2, 203--232 (1997; Zbl 0886.15024)]. In this article, the moments\N\[\NM^{r}_{2p,N} = \sum_{i}x_{i}^{2p} ~ \text{ and } ~M^{c}_{2p, N} = \sum_{i}z_{i}^{2p}\N\]\Nare described in several ways. One way is in terms of a three terms recurrence equation -- building on [\textit{S.-S. Byun}, ``Harer-Zagier type recursion formula for the elliptic GinOE'', Preprint, \url{arXiv:2309.11185}] -- yielding a large \(N\) expansion in terms of power of \(1/N\). Another way is through an exact expression in terms of the \({}_{3}F_{2}\) hypergeometric function. As a byproduct, the authors obtain a three term recurrence for \({}_{3}F_{2}\). Finally, the Stieltjes transform of the density of real eigenvalues is shown to satisfy a differential equation.