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Recently, Kedlaya proves certain formula describing explicitly the Frobeniusstructure on a hypergeometric equation. In this paper, we give a generalizationof it. In our case, the Frobenius matrix is no longer described by p-adic gammafunction, and then we describe it by the p-adic polygamma functions. Since thep-adic polygamma values are linear combinations of p-adic L-values of Dirichletcharacters, it turns out that the Frobenius matrix is described by p-adicL-values. Our result has an application to the study on Frobenius on p-adiccohomology. We show that, for a projective smooth family such that thePicard-Fuchs equation is a hypergeometric equation, the Frobenius matrix on thelog-crystalline cohomology is described by some values of the logarithmicfunction and p-adic L-functions of Dirichlet characters.

Frobenius structure on hypergeometric equations, p-adic polygamma values and p-adic L-values