Frobenius structure and p-adic zeta values

For differential operators of Calabi-Yau type, Candelas, De la Ossa and van Straten conjecture the appearance of p-adic zeta values in the matrix entries of their p-adic Frobenius structure expressed in the standard basis of solutions near a point of maximal unipotent local monodromy. We prove that this phenomenon holds for simplicial and hyperoctahedral families of Calabi-Yau hypersurfaces in n dimensions, in which case the limits of the Frobenius matrix entries are rational linear combinations of products of ${\zeta }_p\left(k\right)$ with $1<k<n$.