  Recently, Kedlaya proves certain formula describing explicitly the Frobenius
structure on a hypergeometric equation. In this paper, we give a generalization
of it. In our case, the Frobenius matrix is no longer described by p-adic gamma
function, and then we describe it by the p-adic polygamma functions. Since the
p-adic polygamma values are linear combinations of p-adic L-values of Dirichlet
characters, it turns out that the Frobenius matrix is described by p-adic
L-values. Our result has an application to the study on Frobenius on p-adic
cohomology. We show that, for a projective smooth family such that the
Picard-Fuchs equation is a hypergeometric equation, the Frobenius matrix on the
log-crystalline cohomology is described by some values of the logarithmic
function and p-adic L-functions of Dirichlet characters.
