Duality of one-variable multiple polylogarithms and their $q$-analogues

TL;DR

This paper presents a new proof of the duality relation for one-variable multiple polylogarithms using connected sums involving hypergeometric functions, and extends the duality to their $q$-analogues.

Contribution

It introduces a novel proof method for duality using connected sums and generalizes the duality to new $q$-analogues of multiple polylogarithms.

Findings

Connected sums involve hypergeometric functions.

Duality relation is extended to $q$-analogues.

New proof method simplifies understanding of duality.

Abstract

The duality relation of one-variable multiple polylogarithms was proved by Hirose, Iwaki, Sato and Tasaka by means of iterated integrals. In this paper, we give a new proof using the method of connected sums, which was recently invented by Seki and the author. Interestingly, the connected sum involves the hypergeometric function in its connector. Moreover, we introduce two kinds of $q$-analogues of the one-variable multiple polylogarithms and generalize the duality to them.