Monodromy of the generalized hypergeometric equation in the maximally unipotent case

Abstract

We consider monodromy groups of the generalized hypergeometric equation

z(θ − α1 ) · · · (θ − αn ) − (θ + β1 − 1) · · · (θ + βn − 1) f (z) = 0 where θ = zd/dz.

We pay particular attention to the maximally unipotent case, where β1 = . . . = βn = 1, and present a theorem that enables us to determine the form of the corresponding monodromy matrices in a suitable basis in the case where (X − e^−2πiα1 ) · · · (X − e^−2πiαn ) is a product of cyclotomic polynomials. A similar result is obtained for general α1 , . . . , αn ∈ Q, where the entries of the monodromy matrices are expressed with the Hurwitz zeta function. In particular, this result gives us an idea of the transcendental numbers to be found in the monodromy matrices.