General evaluation of the two-point integral transform in the Pfaffian setting

Determine explicit closed-form evaluations or broadly applicable structural characterizations for the integral transform [\mathcal{K}_{\vec{\chi},\alpha} f](u,v)=\oint_{\mathcal{C}}\frac{d\zeta\, f(\zeta)}{2\pi i}\exp\big(\frac{u}{\zeta-\chi}+\frac{v}{\zeta-\xi}\big)\frac{\zeta^{2k-\alpha}}{(\zeta-\chi)^{k}(\zeta-\xi)^{k}} that arises in Corollary (factoring derivatives for Pfaffians), for general analytic test functions f and without resorting to case-by-case choices of f.

Background

In the Pfaffian framework for mixed derivatives (symplectic/orthogonal classes), the authors obtain an explicit formula involving an integral transform (equation (transform2)). While the transform can be computed for specific f, a general, explicit evaluation is not available.

A general solution would streamline applications of the main theorems to a wide range of ensembles and kernels where only analyticity assumptions on f are imposed.