Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Elliptic Polylogarithms and Basic Hypergeometric Functions

Giampiero Passarino

TL;DR

The paper builds a computable framework linking elliptic polylogarithms to basic hypergeometric functions, enabling high-precision numerical evaluation through $q$-difference equations and $q$-contiguous relations. It defines EPs $ELi_{n;m}$, derives their analytic continuation, and demonstrates explicit depth-1 and depth-2 constructions, including representations via ${}_2\phi_1$, mixed hypergeometric series, and Barnes contour integrals. Higher-depth EPs are addressed by pole isolation and decompositions into cut and regular parts, with connections to Eisenstein–Kronecker series highlighting the modular structure. This methodology provides practical tools for evaluating elliptic-generalized polylogarithms that appear in challenging Feynman integrals.

Abstract

Multiple elliptic polylogarithms can be written as (multiple) integrals of products of basic hypergeometric functions. The latter are computable, to arbitrary precision, using a q-difference equation and q-contiguous relations.

Elliptic Polylogarithms and Basic Hypergeometric Functions

TL;DR

The paper builds a computable framework linking elliptic polylogarithms to basic hypergeometric functions, enabling high-precision numerical evaluation through -difference equations and -contiguous relations. It defines EPs , derives their analytic continuation, and demonstrates explicit depth-1 and depth-2 constructions, including representations via , mixed hypergeometric series, and Barnes contour integrals. Higher-depth EPs are addressed by pole isolation and decompositions into cut and regular parts, with connections to Eisenstein–Kronecker series highlighting the modular structure. This methodology provides practical tools for evaluating elliptic-generalized polylogarithms that appear in challenging Feynman integrals.

Abstract

Multiple elliptic polylogarithms can be written as (multiple) integrals of products of basic hypergeometric functions. The latter are computable, to arbitrary precision, using a q-difference equation and q-contiguous relations.

Paper Structure

This paper contains 12 sections, 1 theorem, 77 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Elliptic polylogarithms
  3. Basic hypergeometric functions
  4. q-contiguous relations
  5. Analytic continuation
  6. Basic hypergeometric equation
  7. Poles
  8. Elliptic polylogarithms of higher depth
  9. Mixed hypergeometric series
  10. Barnes contour integrals
  11. Eisenstein-Kronecker series
  12. Conclusions

Key Result

Theorem 3.1

Let $f({\mathrm{z}})$ be a continuous function defined for $| {\mathrm{z}} | < r$ and $d \ge 2$ an integer. Suppose that with $a_n(0) = {\mathrm{w}}_n$. Suppose that exists a real number ${\mathrm{M}} > 0$ such that and The $f({\mathrm{z}})$ is uniquely determined by $f(0)$ and the functions $a_n({\mathrm{z}})$.

Figures (3)

  • Figure 1: Behavior of $\upPhi\left( x\,,\,y\,;\,q \right)$, defined in Eq.(\ref{['bphi']}), around the poles at $x = 1/q$ (blue curve) and $x = 1/q^2$ (red curve).
  • Figure 2: Behavior of $\upPhi\left( x\,,\,y\,;\,q \right)$, defined in Eq.(\ref{['bphi']}), as a function of $x$ for $y= 0.1$ and $q= 0.9 + i\,0.04$.
  • Figure 3: The function $\mathrm{eli}_{1\,;\,0}(x\,,\,y\,;\,q)$, Eq.(\ref{['seli']}), for different values of $y$ and $q$ and $1/q_{{\mathrm{r}}} < x < 1/q^2_{{\mathrm{r}}}$, $q_{{\mathrm{r}}} = \mathop{\mathrm{Re}}\nolimits\,q$.

Theorems & Definitions (1)

  • Theorem 3.1: Chen, Hou, Mu