A $q$-analogue of Boyadzhiev-Mneimneh-type binomial sums of finite multi-polylogarithms

TL;DR

This paper introduces a $q$-analogue formula for binomial sums of finite multi-polylogarithms, connecting to known identities as $q$ approaches 1, and extends to sums related to the Cauchy binomial theorem.

Contribution

It provides a new $q$-analogue formula for binomial sums of multi-polylogarithms, generalizing existing identities and linking them to classical results.

Findings

Derived a $q$-analogue formula reducing to Sakugawa-Seki identities as $q\to 1$

Extended the formula to sums related to the Cauchy binomial theorem

Established connections between $q$-analogues and classical polylogarithm identities

Abstract

We give a formula for a $q$-analogue of Boyadzhiev-Mneimneh-type binomial sums of finite multi-polylogarithms. In the limit as $q\to 1$, this formula reduces to an identity equivalent to the Sakugawa-Seki identities. We also give a formula for Boyadzhiev-Mneimneh-type sums corresponding to the Cauchy binomial theorem.