Frobenius map and the p-adic Gamma function

In this note we study the relationship between the power series expansion of the Dwork exponential and the Mahler expansion of the p-adic Gamma function. We exploit this relationship to prove that certain quantities that appeared in our computations in Shapiro (2009) 12] can be expressed in terms of the derivatives of the p-adic Gamma function at 0. We then use this to prove the claim made in Shapiro (2009) 12] about the non-trivial off-diagonal entry in the Frobenius matrix of the mirror quintic threefold.