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

TL;DR

The paper develops a $q$-analogue of binomial sums involving finite multi-polylogarithms, defining $l_k^{\star,q}$ and the binomial-sums $M_n^{q}(\boldsymbol{s}, \boldsymbol{t}; x,y)$. It proves a main closed-form expression (Theorem mainthm) as an explicit nested sum over indices $n_1\ge\cdots\ge n_w$, with a detailed product structure in terms of $[n_i]$, $y$-powers, and $(t_rx+y)$-type factors, and shows the $q\to 1$ limit recovers the classical $l_k^{\star}$-sum identities, including Genčev's and Boyadzhiev–Mneimneh's results. The work also establishes a Cauchy-binomial analogue for these sums, linking to the $[n]$-binomial framework and enabling substitutions that recover the main theorem in this setting. Moreover, it connects the derived identities to Sakugawa–Seki identities via binomial inversion, and discusses how Bradley-type identities emerge as corollaries under particular parameter choices. Overall, the study unifies $q$-calculus for finite multi-polylogarithms with known combinatorial identities and provides a versatile toolkit for finite analogues of polylogarithmic sums.

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.

Theorems & Definitions (19)

• Theorem 1.1: G
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• Remark 2.3
• proof : Proof of Lemma \ref{['lemma:induction_lemma']}
• proof : Proof of Theorem \ref{['thm:mainthm']}