The Widom–Arinkin–Borodin determinant formula

Let $J$ be a jump matrix with factorization $J=\Phi_-^{-1}\Phi_+$, let $\tau_W[J]$ be the Widom determinant, let $\tau_{AB}[J]$ be the corresponding Arinkin–Borodin tau function, and let $\mathbb{I}$ denote the identity jump. Widom–Arinkin–Borodin formula. The Widom determinant has the expression

$$\tau_W[J]=\frac{\tau_{AB}[J]\tau_{AB}[\mathbb{I}]}{\tau_{AB}[\Phi_+]\tau_{AB}[\Phi_-^{-1}]}.$$

This statement is conditional on defining the Arinkin–Borodin tau function for arbitrary jumps in the relevant equivalence class, so its status is unresolved in the supplied text.