Elliptic polylogarithms and iterated integrals on elliptic curves I: general formalism

TL;DR

This paper introduces a new class of iterated integrals on elliptic curves that generalize multiple polylogarithms, with potential applications in high-energy physics calculations.

Contribution

It develops a formalism for elliptic iterated integrals with simple poles, connecting them to existing elliptic polylogarithms and enabling practical computations.

Findings

The integrals span the same function space as multiple elliptic polylogarithms.

They have at most logarithmic singularities.

Demonstrated application in hypergeometric function expansions.

Abstract

We introduce a class of iterated integrals, defined through a set of linearly independent integration kernels on elliptic curves. As a direct generalisation of multiple polylogarithms, we construct our set of integration kernels ensuring that they have at most simple poles, implying that the iterated integrals have at most logarithmic singularities. We study the properties of our iterated integrals and their relationship to the multiple elliptic polylogarithms from the mathematics literature. On the one hand, we find that our iterated integrals span essentially the same space of functions as the multiple elliptic polylogarithms. On the other, our formulation allows for a more direct use to solve a large variety of problems in high-energy physics. We demonstrate the use of our functions in the evaluation of the Laurent expansion of some hypergeometric functions for values of the indices close to half integers.