[Paper Review] On $q$-Analogs of Some Families of Multiple Harmonic Sum and Multiple Zeta Star Value Identities

Khodabakhsh Hessami Pilehrood, Tatiana Hessami Pilehrood+1 more|arXiv (Cornell University)|Jul 30, 2013

Advanced Mathematical Identities13 references10 citations

TL;DR

This paper establishes $q$-analog identities for multiple zeta star values (MZSV) and multiple harmonic sums (MHS), generalizing classical duality relations such as the Two-one formula. By introducing a $q$-deformation of Euler sums and using O-plus operations, the authors derive $q$-analog duality theorems that recover classical identities in the limit $q \to 1$, proving a conjecture by Ihara et al. on the spanning of MZV spaces by Hoffman $\star$-elements with weights in \{2,3\}.

ABSTRACT

In recent years, there has been intensive research on the ${\mathbb Q}$-linear relations between multiple zeta (star) values. In this paper, we prove many families of identities involving the $q$-analog of these values, from which we can always recover the corresponding classical identities by taking $q o 1$. The main result of the paper is the duality relations between multiple zeta star values and Euler sums and their $q$-analogs, which are generalizations of the Two-one formula and some multiple harmonic sum identities and their $q$-analogs proved by the authors recently. Such duality relations lead to a proof of the conjecture by Ihara et al. that the Hoffman $\star$-elements $ζ^{\star}(s_1,\dots,s_r)$ with $s_i\in\{2,3\}$ span the vector space generated by multiple zeta values over ${\mathbb Q}$.

Motivation & Objective

• To extend classical identities involving multiple zeta star values (MZSV) and Euler sums to their $q$-analogues.
• To establish a $q$-deformed duality between $q$-MZSV and $q$-Euler sums, generalizing the Two-one formula.
• To prove that the Hoffman $\star$-elements $\zeta^\star(s_1,\dots,s_r)$ with $s_i \in \{2,3\}$ span the rational vector space of multiple zeta values.
• To provide a systematic framework for generating $q$-analog identities using O-plus operations and signed index structures.
• To recover previously unproven classical MZSV identities by taking the limit $q \to 1$ in new $q$-identities.

Proposed method

• Introduces a signed index set $\mathbb{D} = \mathbb{N} \cup \overline{\mathbb{N}}$ with O-plus ($\oplus$) operation to unify alternating and non-alternating sums.
• Defines $q$-analogues of multiple harmonic sums ($H_n^\star$) and multiple zeta star values ($\zeta^\star$) using $q$-integers $[k]_q$ and signed exponents.
• Applies attaching rules (e.g., (19) and (20)) to recursively build $q$-analog identities by modifying index strings via $\oplus$ and sign transformations.
• Uses the $q$-Euler sum $\mathfrak{z}^\sharp$-function to express $q$-MZSVs in terms of $q$-analog Euler sums with transformed index sequences.
• Applies the limit $q \to 1$ to $q$-identities to recover known classical identities, including the Two-one formula and new MZSV identities.
• Employs generating functions and $q$-series identities (e.g., Lemma 2.3) to verify convergence and algebraic structure in the $q \to 1$ limit.

Experimental results

Research questions

• RQ1 Can classical duality identities between multiple zeta star values and Euler sums be generalized to $q$-analogs?
• RQ2 What is the structure of $q$-analog Euler sums that correspond to $q$-MZSVs under the O-plus operation?
• RQ3 Do the $q$-analog identities recover known classical identities in the limit $q \to 1$?
• RQ4 Can the $q$-deformed duality relations be used to prove that Hoffman $\star$-elements with weights in $\{2,3\}$ span the rational vector space of MZVs?
• RQ5 What is the precise form of $q$-analog identities for MZSVs of the form $\zeta^\star(\{2\}^a, 1, \{2\}^b, 3, \{2\}^c, 1)$?

Key findings

• The paper proves that $\zeta^\star(\{2\}^a, {\bf s}) = \iota_{\bf s} \zeta^\sharp(2a \oplus \lambda_1, \dots, \lambda_m)$, generalizing the Two-one formula to $q$-analogs.
• For any $q$-MZSV of the form $\zeta^\star(\{2\}^a, 1, \{2\}^b, 1, \{2\}^c, 3, \{2\}^d, 1)$, the identity $\zeta^\star[\{2\}^a,1,\{2\}^b,1,\{2\}^c,3,\{2\}^d,1] = \sum \mathfrak{z}[\cdot]$ holds with seven distinct $q$-Euler sum terms.
• The $q$-analog identity for $\zeta^\star[\{2\}^a,3,\{2\}^b,1] + \zeta^\star[\{2\}^b,3,\{2\}^a,1] - \zeta^\star[\{2\}^{a+1}] olimits \zeta^\star[\{2\}^{b+1}] = (1-q)\mathfrak{z}[2a+2b+3;a+b+2;2]$ is established, with the correction term vanishing as $q \to 1$.
• By taking $q \to 1$, the paper recovers and proves a previously unproven classical MZSV identity (3) from the introduction, confirming its validity.
• The authors prove that the Hoffman $\star$-elements $\zeta^\star(s_1,\dots,s_r)$ with $s_i \in \{2,3\}$ span the rational vector space of multiple zeta values, confirming a conjecture by Ihara et al.
• The duality between $q$-MZSVs and $q$-Euler sums is preserved under recursive application of attaching rules, enabling systematic generation of new identities.