Convolutions on the complex torus

TL;DR

This paper explores the algebraic structures of quasi-elliptic functions on the complex torus, comparing multiplication and convolution, and relates these to recent developments in Feynman integral computations.

Contribution

It explicitly computes the map between bases for the ring structures of quasi-elliptic functions and connects these to polynomial sequences linked to Eulerian polynomials.

Findings

Derived the basis transformation map for quasi-elliptic functions

Linked algebraic structures to polynomial sequences and Feynman integrals

Provided insights into the algebraic properties of functions on the complex torus

Abstract

``Quasi-elliptic'' functions can be given a ring structure in two different ways, using either ordinary multiplication, or convolution. The map between the corresponding standard bases is calculated. A related structure has appeared recently in the computation of Feynman integrals. The two approaches are related by a sequence of polynomials closely tied to the Eulerian polynomials.