Logarithmic $A$-hypergeometric series ${\textrm I}\! {\textrm I}\! {\textrm I}$
Go Okuyama, Mutsumi Saito
TL;DR
The paper develops a comprehensive extension of Frobenius's method for $A$-hypergeometric systems, proving that all $A$-hypergeometric series can be constructed from perturbations of exponents. It builds a duality framework between formal power series and differential operators, introduces minimal weight and fake-exponent concepts, and derives precise indicial-equations that govern coefficient polynomials. By extending Frobenius perturbation to the lattice $L$ and presenting an $L$-perturbation theory with clear sufficiency conditions, the work unifies the construction of logarithmic series and clarifies when extended perturbations are needed. The inclusion of explicit examples and an open question about the necessity of the extended method highlights practical steps for applying and testing the framework in diverse $A$-hypergeometric settings.
Abstract
This paper is the third in a series exploring Frobenius's method for $A$-hypergeometric systems. Frobenius's method is a classical technique for constructing logarithmic series solutions of differential equations by perturbing exponents of generic series solutions. We show that all $A$-hypergeometric series solutions can be obtained via this method. Building upon our prior studies, we develop a duality framework between formal power series and differential operators, introduce minimal vectors with respect to a generic weight, and establish key results on logarithmic coefficients of $A$-hypergeometric series. We extend Frobenius's method and prove its sufficiency in constructing all $A$-hypergeometric series solutions. Furthermore, we explore conditions under which the Frobenius method developed in our previous studies suffices and we pose an open question on the necessity of the extended one.