Monodromy of the generalized hypergeometric equation in the Frobenius basis

Abstract

We consider monodromy groups of the generalized hypergeometric equation \begin{equation*} \big[z(\theta+\alpha_{1})\cdots (\theta+\alpha_{n})-(\theta+\beta_{1}-1)\cdots (\theta+\beta_{n}-1)\big]f(z) = 0\text{, where }\theta = z d/dz, \end{equation*} in a suitable basis, closely related to the Frobenius basis. We pay particular attention to the maximally unipotent case, where $\beta_{1}=\ldots=\beta_{n}=1$, and present a theorem that enables us to determine the form of the corresponding monodromy matrices in the case where $(X-e^{-2\pi i\alpha_{1}})\cdots (X-e^{-2\pi i\alpha_{n}})$ is a product of cyclotomic polynomials.