  The moments of the real eigenvalues of real Ginibre matrices are investigated
from the viewpoint of explicit formulas, differential and difference equations,
and large $N$ expansions. These topics are inter-related. For example, a third
order differential equation can be derived for the density of the real
eigenvalues, and this can be used to deduce a second order difference equation
for the general complex moments $M_{2p}^{\rm r}$. The latter are expressed in
terms of the ${}_3 F_2$ hypergeometric functions, with a simplification to the
${}_2 F_1$ hypergeometric function possible for $p=0$ and $p=1$, allowing for
the large $N$ expansion of these moments to be obtained. The large $N$
expansion involves both integer and half integer powers of $1/N$. The three
term recurrence then provides the large $N$ expansion of the full sequence $\{
M_{2p}^{\rm r} \}_{p=0}^\infty$. Fourth and third order linear differential
equations are obtained for the moment generating function and for the Stieltjes
transform of the real density, respectively, and properties of the large $N$
expansion of these quantities are determined.
