Sums of Two-Parameter Deformations of Multiple Polylogarithms

Abstract

In this paper, we introduce a generating function of sums of two-parameter deformations of multiple polylogarithms, denoted by Φ₂(a;p,q), and study a q-difference equation satisfied by it. We show that this q-difference equation can be solved by expanding Φ₂(a;p,q) into power series of the parameter p and then using the method of variation of constants. By letting \(p \rightarrow 0\) in the main theorem, we find that the generating function of sums of q-interpolated multiple zeta values can be written in terms of the q-hypergeometric function ₃ϕ₂, which is due to Li-Wakabayashi.