Monodromy of the generalized hypergeometric equation in the Frobenius basis

TL;DR

This paper investigates the monodromy groups of generalized hypergeometric equations, especially in the maximally unipotent case, providing explicit forms of monodromy matrices when certain polynomial conditions are met.

Contribution

It introduces a theorem that determines the monodromy matrices for hypergeometric equations with cyclotomic polynomial conditions, focusing on the Frobenius basis and unipotent cases.

Findings

Explicit monodromy matrix forms derived for specific polynomial cases

Characterization of monodromy groups in the maximally unipotent case

Connection between polynomial factorization and monodromy structure

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.