Finite $q$-multiple harmonic sums on $1-\cdots-1,A,1-\cdots-1$ indices

TL;DR

This work extends explicit expressions for finite $q$-multiple harmonic sums on indices with a single $A$-block among ones to general $A\ge3$, building on prior results for $A=1,2$. By leveraging elementary symmetric functions and degenerate Bernoulli numbers, the authors derive polynomial expressions in $n$ and $m$ for sums involving $A=2,3,4,5$, and provide degenerate-Bernoulli representations that unify these results. They also analyze the more intricate $2- frac{1}{2}- obreak 2,A,2- obreak 2$ configurations, giving explicit odd/even-$A$ formulas and concrete examples, and outline a program for extending to higher exponents. The results advance explicit computability of these finite $q$-harmonic sums, with potential implications for $q$-zeta values and related combinatorial structures. Overall, the paper contributes new closed forms and a flexible framework (symmetric-function and degenerate Bernoulli approaches) for evaluating nonuniform index patterns in finite $q$-MZV-like sums.

Abstract

In this paper, we give explicit expressions about $q$-harmonic sums on $1-\cdots-1,A,1-\cdots-1$ indices. When $A=1$, many previous authors have studied and showed the identities, expressions, and properties. There are many results for explicit expressions about $q$-multiple zeta values or $q$-harmonic sums on $A-\cdots-A$ indices. Though there is the way to treat $q$-multiple zeta values unless the indices are the same, it has been successful to get the explicit expression of $q$-harmonic sums on $1-\cdots-1,A,1-\cdots-1$ indices when $A=2$. In this paper, we shall consider more general results when $A\ge 3$.

Theorems & Definitions (11)

• Lemma 1
• Theorem 1
• proof
• Theorem 2
• Theorem 3
• Lemma 2
• proof
• proof : Proof of Theorem \ref{['th:mzv1s-degberpol']}.
• Lemma 3
• Theorem 4