Simultaneous analytic continuation in k and h for Ginibre mixed-derivative moments

Establish a simultaneous analytic continuation in the two parameters k and h for the large-N limit of the normalized mixed moment \lim_{N\to\infty}\frac{\langle |D_N(\chi)|^{2h} |D_N(\chi)'|^{2(k-h)}\rangle}{\prod_{j=N}^{N+k-1}\pi j!} in the complex Ginibre ensemble, extending the formula beyond integer k and h without restricting to lines of constant k-h.

Background

For first-derivative moments in the Ginibre ensemble, the authors derive an explicit large-N formula (equation (eq:corGinUEZerothFirstExplicit)). They observe that, with k−h fixed, the expression can be analytically continued in either k or h separately.

However, a joint analytic continuation in both k and h simultaneously remains unresolved. Solving this would parallel established analytic continuation results in related CUE settings and broaden applicability to non-integer parameter regimes.

References

At constant $k-h \in \mathbb{N}_0$, equation eq:corGinUEZerothFirstExplicit can be analytically continued in either $k$ or $h$. The analytic continuation independently in both variables is still an open question.

— Derivative relations for determinants, Pfaffians and characteristic polynomials in random matrix theory  (2603.29510 - Akemann et al., 31 Mar 2026) in Example (Moments with first derivatives), Subsection 3.1 (Example: the complex Ginibre ensemble)