Harer-Zagier type recursion formula for the elliptic GinOE

We consider the real eigenvalues of the elliptic Ginibre matrix indexed by the non-Hermiticity parameter $\tau \in [0,1]$, and present a Harer-Zagier type recursion formula for the even moments in the form of an $11$-term recurrence relation. For the symmetric GOE case ($\tau=1$), it reduces to a known 5-term recurrence relation. On the other hand, for the asymmetric cases when $\tau < 1$, the recursion formula is new, even in the special case of the well-studied Ginibre ensemble ($\tau=0$), where it reduces to a 3-term recurrence. For the proof, we derive a seventh-order linear differential equation for the moment generating function.