Finite $q$-multiple harmonic sums on $2-\cdots-2,1-\cdots-1$ indices

TL;DR

This work extends finite $q$-multiple zeta values to mixed index patterns, notably $2$-to-$2$ and $1$-to-$1$ configurations. By introducing permutation-sum quantities $\mathcal{Y}_n(s_1,\dots,s_m)$ and employing generating-function techniques tied to roots of unity, the authors derive explicit closed forms for averaged sums like $\mathcal{Y}_n(\underbrace{2,\dots,2}_{r},\underbrace{1,\dots,1}_{m-r})$, namely $\mathcal{Y}_n(2^{r},1^{m-r})=\frac{1}{(r+1)n}\binom{m}{r}\bigl(\binom{n-1}{m}+(-1)^r\binom{n-1}{m+r+1}\bigr)$. They further develop a network of product identities linking mixed-index sums to $\mathcal{Y}_n$ terms with $3$’s and higher, and obtain parity-based results for products with the same parity of indices, including a key identity $\displaystyle \sum_{j=0}^{n-m-1}\binom{m-\ell+2j}{j}\mathcal{Y}_n(\underbrace{2,\dots,2}_{\ell-j},\underbrace{1,\dots,1}_{m-\ell+2j})=\frac{1}{(m+1)(\ell+1)}\binom{n-1}{m}\binom{n-1}{\ell}$. The paper also treats negative powers, providing explicit formulas for $\mathfrak Z_n(\zeta_n;;\underbrace{-1,\dots,-1}_{\ell})$ and $\mathfrak Z_n(\zeta_n;;\underbrace{-2,\dots,-2}_{\ell})$, and demonstrates how these results yield broader identities for finite $q$-MZVs with nonuniform indices. Overall, the results give concrete polynomial expressions for averaged finite $q$-MZVs in mixed-index cases and connect to prior uniform-index results, while outlining challenges in obtaining termwise explicit formulas due to imaginary parts.

Abstract

There are many results for explicit expressions about $q$-multiple zeta values or $q$-harmonic sums on $A-\cdots-A$ indices, that is, the indices are the same. Though 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 $1-\cdots-1,2,1-\cdots-1$ indices. In this paper, we shall consider more general results when the ratio of indices of $2$ to indices of $1$ increases.

Theorems & Definitions (21)

• Theorem 1
• proof
• Theorem 2
• Proposition 1
• proof
• proof : Proof of Theorem \ref{['th:5']}.
• Corollary 1
• Theorem 3
• Proposition 2
• proof