Confluence relations for $q$-analogues of multiple zeta values

Abstract

In this paper, we construct confluence relations for $q$-analogues of multiple zeta values by adapting the classical construction to the $q$-setting. Our construction uses $q$-analogues of multiple polylogarithms depending on an auxiliary variable $z$, functional relations arising from their $q$-differential equations, and the (regularized) limit as $z$ tends to $1$. We prove that the resulting family of relations contains the stuffle product relations and, assuming a certain conjectural identity, the duality relations for both the Bradley--Zhao and Schlesinger--Zudilin models.