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

## Abstract

We introduce a class of iterated integrals, defined through a set of linearly independent integration kernels on elliptic curves. As a direct generalization 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. As a result, 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.

- Authors:

- Humboldt-Univ. zu Berlin, Berlin (Germany)
- European Organization for Nuclear Research (CERN), Geneva (Switzerland); Univ. Catholique de Louvain, Louvain-La-Neuve (Belgium)
- Stanford Univ., Stanford, CA (United States)
- European Organization for Nuclear Research (CERN), Geneva (Switzerland)

- Publication Date:

- Research Org.:
- SLAC National Accelerator Lab., Menlo Park, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1458523

- Grant/Contract Number:
- AC02-76SF00515

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2018; Journal Issue: 5; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; NLO Computations; QCD Phenomenology

### Citation Formats

```
Broedel, Johannes, Duhr, Claude, Dulat, Falko, and Tancredi, Lorenzo.
```*Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism*. United States: N. p., 2018.
Web. doi:10.1007/jhep05(2018)093.

```
Broedel, Johannes, Duhr, Claude, Dulat, Falko, & Tancredi, Lorenzo.
```*Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism*. United States. doi:10.1007/jhep05(2018)093.

```
Broedel, Johannes, Duhr, Claude, Dulat, Falko, and Tancredi, Lorenzo. Tue .
"Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism". United States.
doi:10.1007/jhep05(2018)093. https://www.osti.gov/servlets/purl/1458523.
```

```
@article{osti_1458523,
```

title = {Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism},

author = {Broedel, Johannes and Duhr, Claude and Dulat, Falko and Tancredi, Lorenzo},

abstractNote = {We introduce a class of iterated integrals, defined through a set of linearly independent integration kernels on elliptic curves. As a direct generalization 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. As a result, 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.},

doi = {10.1007/jhep05(2018)093},

journal = {Journal of High Energy Physics (Online)},

number = 5,

volume = 2018,

place = {United States},

year = {Tue May 15 00:00:00 EDT 2018},

month = {Tue May 15 00:00:00 EDT 2018}

}