Interpolated Multiple Zeta Values

• Interpolated multiple zeta values are a one-parameter family of functions that continuously bridge strict MZVs and star MZSVs via t-weighted merging of indices.
• They maintain deformed shuffle and stuffle relations, extending classical double shuffle and Ohno-type identities to a unifying algebraic framework.
• Their polynomial formulation enables generalizations to finite fields, q-analogues, and Schur-type extensions, offering new insights in number theory and combinatorics.

Interpolated multiple zeta values (IMZVs) form a one-parameter family of $\mathbb{R}[t]$-valued functions that continuously interpolate between ordinary multiple zeta values (MZVs) and multiple zeta-star values (MZSVs). Introduced by Yamamoto, IMZVs capture the full combinatorial algebraic structure of both strict (MZV) and non-strict (MZSV) sum cases, providing a unifying framework for their algebraic relations, sum formulas, duality properties, and generalized shuffle/stuffle products. Their polynomial nature enables generalizations to finite fields, function fields, $q$-analogues, and Schur-type generalizations, and exposes deep connections with hypergeometric generating functions, harmonic algebras, and combinatorics of multisets and partitions.

1. Definition and Interpolation Principle

Let $k = (k_1, ..., k_r)$ be an admissible index (each $k_i\in\mathbb{N}$, $k_r \ge 2$). The interpolated multiple zeta value $\zeta^t(k)\in\mathbb{R}[t]$ is defined by summing over all ways of merging adjacent $k_i$ by either a comma or a "+" sign, with each "+" counted with a factor $t$: $$\zeta^t(k_1,\ldots, k_r) = \sum_{\sigma\in S} t^{\#\{i: \sigma_i=+\}} \zeta(k(\sigma)),$$ where $$S = \{\sigma_1, ..., \sigma_{r-1}: \sigma_i\in\{',', '+'\}\}$$, and $k(\sigma)$ is the index formed by merging adjacent entries according to $\sigma$ (Hirose et al., 2021).

This construction yields

$$\zeta^0(k_1, ..., k_r) = \zeta(k_1, ..., k_r) \quad \text{(strict MZV)},$$

$$\zeta^1(k_1, ..., k_r) = \zeta^{*}(k_1, ..., k_r) \quad \text{(star MZV)}.$$

Thus, $\zeta^t$ interpolates between both variants for all integer indices and polynomially in $t$.

2. Algebraic Structure and Ohno-type Relations

A central algebraic property of IMZVs is their compatibility with the extended double shuffle and harmonic (stuffle) products, defined in non-commutative polynomial algebras generated by (ordered) words in $x, y$ (where $z_k = x^{k-1} y$). The $t$-shuffle and $t$-stuffle products deform the standard shuffle and stuffle relations by explicit $t$-dependent coefficients (Li et al., 2016, Li, 2019, Sarkar et al., 2024):

• $t$-stuffle:

$$z_{k_1}\cdots z_{k_n} *_t z_{\ell_1}\cdots z_{\ell_m} = \sum\text{(all stufflings)} (1-2t)^{\#\text{equal-merged}} (t^2-t)^{\#\text{unequal-merged}} \ ...$$

• $t$-shuffle (similarly defined via recursion with $t$-weighted corrections).

This structure leads to a direct analog of the Ohno relation, one of the most symmetric and far-reaching equalities in the theory of MZVs. For an index $k$ and its Hoffman dual $k^*$,

$$\sum_{e_1+...+e_r=m} \zeta^t(k_1+e_1, ..., k_r+e_r) = \sum_{e_1'+...+e_{r^*}'=m} \zeta^t(k_1^* + e_1', ..., k_{r^*}^* + e_{r^*}'),$$

as $\mathbb{R}[t]$-valued polynomials for all $m \ge 0$ (Hirose et al., 2021). This encompasses the Ohno relation for both strict and star MZVs as special cases.

3. Sum Formulas, Generating Functions, and Combinatorial Aspects

The sum-formula for IMZVs, a generalization of the classical MZV sum formula, is derived via this Ohno-type identity: $$\sum_{\substack{k_1+\dots+k_r=k \ k_i\ge 1,\, k_r\ge 2}} \zeta^t(k_1,\ldots,k_r) = \left(\sum_{j=0}^{r-1} \binom{k-1}{j} t^j (1-t)^{r-1-j}\right) \zeta(k)$$ for $k>r\ge1$ (Hirose et al., 2021).

Generating functions for sums with given weight, depth, and height are representable in hypergeometric terms. For example, for fixed weight, depth, and height, the corresponding sums are governed by ${}_3F_2$-series (Li et al., 2016). For indices built from constant blocks, the generating functions admit explicit Euler-product and Bell polynomial expansions, which interpolate between symmetric and full homogeneous polynomials (Kuba, 2019, Kuba, 2020).

The combinatorial and probabilistic interpretations (e.g., via weighted multisets, random compositions) reveal limit laws and the deep connection between IMZVs and symmetric function theory (Kuba, 2019).

4. Extensions: Multi-parameter, Schur, Polynomial, and Finite Field Generalizations

The interpolation principle admits several significant generalizations:

• Multi-interpolated MZVs: Incorporate a sequence of parameters $\vec{t} = (t_1, t_2, ...)$, assigning a distinct weight to each equality between adjacent indices at level $j$:

$$\zeta^{\vec{t}}(i_1,\ldots,i_k) = \sum_{\ell_1\ge \cdots \ge \ell_k \ge 1} \prod_{j\ge 1} t_j^{\sigma_j(\ell_1, ..., \ell_k)} \prod_{m=1}^k \ell_m^{-i_m},$$

where $\sigma_j$ counts the number of adjacent equalities at value $j$ (Kuba, 2020).

• Schur-type IMZVs: IMZVs are extended to fillings of Young diagrams, weighting vertical and horizontal equalities differently, and the Jacobi-Trudi determinant formula holds for the Schur generalization (Bachmann, 2017, Nakasuji et al., 2022).
• Polynomial and Symmetric IMZVs: Interpolated polynomial MZVs $\zeta_{x,y}^t$ generalize both symmetric and star variants, capturing all interpolation schemes as specializations (Hirose et al., 2022).
• Finite Field and $q$-Analogues: There is a rich theory of finite field analogues (finite multiple zeta values), $q$-analogues, and $t$-$q$ versions, with all major algebraic relations (shuffle, stuffle, duality, cyclic sum) preserved (Li et al., 2017, Wakabayashi, 2016, Murahara et al., 2019, Shen, 2019).

5. Functional Equations, Decomposition Formulas, and Explicit Evaluations

The IMZV algebra admits explicit generalized Euler decomposition formulas, providing $t$-dependent multivariate polynomial expansions for products of IMZVs (Sarkar et al., 31 Jan 2026): $$\zeta^t(\mathbf{l}) \cdot \zeta^t(\mathbf{m}) = \sum_{\alpha} C(\alpha, t) \left[ \zeta^t(\alpha+1) - t \zeta^t(\alpha'+1)\right] + ...,$$ with $t=0$ and $t=1$ yielding the classical Euler decomposition for MZVs and MZSVs.

Evaluation formulas for special families—such as $\zeta^t(\{2\}^n,3)$, and for blocks involving only 1, 2, and 3—are obtainable via harmonic algebra computations and generating-function techniques. Recent work yields closed-form single zeta polynomial evaluations via creative telescoping and hypergeometric reductions (Au et al., 2024, Li et al., 2024).

A selection of explicit examples includes:

• $\zeta^t(2,2) = 2t^2-2t$ (Li, 2019)
• $\zeta^t(1,3)+\zeta^t(2,2) = (1+2t)\zeta(4)$ (Hirose et al., 2021) These allow interpolation of classical formulas and provide a test bed for conjectures on relations among higher-weight IMZVs.

6. Open Questions and Research Directions

Recent developments highlight several avenues for further exploration:

• Combinatorial and algebraic bases: Determination of $\mathbb{Q}[t]$-bases for IMZVs remains open, as does the extension of Broadhurst-Kreimer-type counting to the interpolated setting (Li, 2019).
• Dimensional conjectures and structural classification: The full linear/algebraic polynomial structure over $\mathbb{Q}[t]$ is yet to be fully resolved, especially for multi-parameter and Schur-type generalizations (Kuba, 2020, Bachmann, 2017).
• Explicit evaluations beyond height-one: While broad classes of IMZVs are reducible to polynomials in Riemann zeta values for certain admissible indices, the structure for general blocks or higher height (in terms of 1,2,3,...) remains elusive (Li et al., 2024).
• Connections to poly-Bernoulli zeta and motivic lifts: IMZVs bridge multiple zeta values with poly-Bernoulli numbers and encode deep analytic and algebraic relations (Kaneko et al., 2015).
• $q$-parameter and function field generalizations: Analysis of $t$-$q$-MZVs, finite field analogues, and their algebraic relations continues to be a fertile area for new algebraic and analytic phenomena (Shen, 2019, Li et al., 2017).

The IMZV framework thus serves as a central organizing object in the modern study of iterated sums, special function theory, and arithmetic geometry, interconnecting a multitude of classical and contemporary mathematical structures.