A new kind of automorphic form and a proof of the essential transformation laws

TL;DR

This paper introduces a novel vector automorphic form related to Hecke triangle groups and proves the associated functional equations, expanding the understanding of automorphic forms and their transformation properties.

Contribution

It defines a new vector analogue of automorphic forms over Hecke triangle groups and proves their essential transformation laws.

Findings

Established a new vector automorphic form over Hecke triangle groups

Proved the functional equations for these forms modulo group generators

Extended the theory of automorphic forms to a broader class of groups

Abstract

We utilize the structure of quasiautomorphic forms over an arbitrary Hecke triangle group to define a new vector analogue of an automorphic form. We supply a proof of the functional equations that hold for these functions modulo the group generators.