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In this note we study the relationship between the power series expansion ofthe Dwork exponential and the Mahler expansion of the $p$-adic Gamma function.We exploit this relationship to prove that certain quantities that appeared inour previous computations of the Frobenius map can be expressed in terms of thederivatives of the $p$-adic Gamma function at 0. This is used to prove aconjecture about the non-trivial off-diagonal entry in the Frobenius matrix ofthe mirror quintic threefold.

Frobenius map and the $p$-adic Gamma function