Hecke-Symmetry and Rational Period Functions

TL;DR

This paper advances the understanding of rational period functions on Hecke groups by analyzing Hecke-symmetry of poles, providing new expressions for these functions, and illustrating their properties with examples.

Contribution

It introduces a new expression for rational period functions on Hecke groups considering Hecke-symmetry, and clarifies previous theoretical results.

Findings

Hecke-symmetry of poles influences rational period functions.

New explicit expressions for rational period functions are derived.

Examples illustrate the properties and corrections of earlier theorems.

Abstract

In this paper we continue work in the direction of a characterization of rational period functions on the Hecke groups. We examine the role that Hecke-symmetry of poles plays in this setting, and pay particular attention to non-symmetric irreducible systems of poles for a rational period function. This gives us a new expression for a class of rational period functions of any positive even integer weight on the Hecke groups. We illustrate these properties with examples of specific rational period functions. We also correct the wording of a theorem from an earlier paper.