Dwork congruences via q-deformation

We consider a system of polynomials $T_s(z,q)\in \mathbb{Z}[z,q]$ which appear as truncations of the K-theoretic vertex function for the cotangent bundles over Grassmannians $T^{\ast }Gr(k,n)$. We prove that these polynomials satisfy a natural $q-$deformation of Dwork’s congruences $$\frac{T_{s+1}(z,q)}{T_s(z^p,q^p)}\equiv \frac{T_s(z,q)}{T_{s-1}(z^p,q^p)}\text{ (mod }[p^s{]}_q)$$ In the limit $q\to 1$ we recover the main result of arXiv:2302.03092v3