  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 previous computations of the Frobenius map can be expressed in terms of the
derivatives of the $p$-adic Gamma function at 0. This is used to prove a
conjecture about the non-trivial off-diagonal entry in the Frobenius matrix of
the mirror quintic threefold.
