Multiple t-Harmonic Star Sums

Updated 31 January 2026

Multiple t-harmonic star sums are generalizations of multiple zeta-star values that restrict summation to odd integers with structured star conditions.
They employ advanced generating function techniques to derive explicit combinatorial evaluations, enhancing studies in modular forms and level-N L-values.
Their rich algebraic structure, including stuffle, shuffle, and duality relations, deepens our understanding of analytic number theory and combinatorial identities.

Multiple $t$ -Harmonic Star Sums are generalizations of multiple zeta-star values (MZSVs) in which summation is restricted to odd-integer arguments, and which are equipped with structured “star” (harmonic) summation conditions and often studied via their generating functions and explicit combinatorial evaluations. This class includes objects of central interest in the theory of level- $N$ multiple $L$ -values, modular forms, and algebraic relations among special functions. Their core features and computational apparatus, including analytic continuation, weighted sum formulae, symmetric sums, deep generating function techniques, and explicit evaluations, have made them a focus within modern algebraic and analytic number theory.

1. Definitions and Notational Framework

Let $r \geq 1$ , $\mathbf{s} = (s_1, \ldots, s_r)$ a sequence of positive integers with $s_1 > 1$ . The multiple $t$ -value and its star (harmonic) variant are defined by

$t(s_1, \ldots, s_r) = \sum_{k_1 > \cdots > k_r \geq 1} \frac{1}{(2k_1 - 1)^{s_1} \cdots (2k_r - 1)^{s_r}}$

$t^*(s_1, \ldots, s_r) = \sum_{k_1 \geq \cdots \geq k_r \geq 1} \frac{1}{(2k_1 - 1)^{s_1} \cdots (2k_r - 1)^{s_r}}$

with $t(\emptyset)=t^*(\emptyset)=1$ .

The finite $n$ -truncation, called the multiple $t$ -harmonic star sum, is

$t^*_n(s_1, \ldots, s_r) = \sum_{n \geq k_1 \geq \cdots \geq k_r \geq 1} \frac{1}{(2k_1 - 1)^{s_1} \cdots (2k_r - 1)^{s_r}}$

with $t_n^*(\,\emptyset\,)=1$ , and $t^*_n(\mathbf{s}) \to t^*(\mathbf{s})$ as $n \to \infty$ (Li et al., 2022). For repeated blocks, $\{2\}^a$ denotes the sequence of $2$ repeated $a$ times.

Multiple $t$ -harmonic star sums admit rich algebraic structure, including stuffle product, weighted sums, and interpolated families (Murahara et al., 2019, Li et al., 2022).

2. Generating Functions for Multiple $t$ -Harmonic Star Sums

The main computational leverage arises from generating functions encoding all multiple $t$ -harmonic star sums with certain block structures: $G_n(\mathbf{c}; z_0, \ldots, z_d) = \sum_{a_0, \ldots, a_d \geq 0} t^*_n(\{2\}^{a_0}, c_1, \{2\}^{a_1}, \ldots, c_d, \{2\}^{a_d}) \prod_{j=0}^d z_j^{2a_j}$ with separator indices $\mathbf{c}=(c_1, \dots, c_d)$ , $c_i \ne 2$ , $|z_j|<1$ (Li et al., 2022).

A closed-form expression is given by

$G_n(\mathbf{c}; z) = \sum_{1 \leq k_0 \leq \cdots \leq k_d \leq n} \frac{(-1)^{k_0 + \cdots + k_d} V_n^\#(\{1\}^{c_i - 3}; k_{i-1}, k_i)}{\prod_{i=0}^d ((2k_i - 1)^2 - z_i^2)} \left(\binom{2n}{n} 2^{2n-2}\right)^{-1}$

where $V_n^\#$ is a combinatorial transition term and $d_i = 8c_i + 8c_{i+1}$ , $c_0 = c_{d+1}=1$ .

Specializations yield generating functions for e.g., $t_n^*(\{2\}^a, 3, \{2\}^b, 1, \ldots)$ . Coefficient extraction in the $z_j$ variables provides explicit finite-sum expressions for specific block patterns (Li et al., 2022).

3. Limit Transition and Explicit Star Value Formulas

Taking $n \to \infty$ gives the generating function for infinite star-values: $G(\mathbf{c}; z) = \sum_{a_0, \ldots, a_d \geq 0} t^*(\{2\}^{a_0}, c_1, \ldots, c_d, \{2\}^{a_d}) \prod_{j=0}^d z_j^{2a_j}$ satisfying analogous sum and convergence properties.

Explicit evaluation for index patterns with multiple blocks of twos and interpolated indices is given by

$t^*(\{2\}^{a_0}, c_1, \ldots, c_d, \{2\}^{a_d}) = \sum_{1 \leq k_0 \leq \cdots \leq k_d} (-1)^{k_0 + \cdots + k_d} \prod_{i=0}^d (2k_i - 1)^{2a_i-d_i + 3{\mathbb{1}}_{i>0}} V^\#(\{1\}^{c_i-3}; k_{i-1}, k_i)$

with the indicator for blocks beyond the initial one and the same $d_i$ as before (Li et al., 2022).

One-block constraints and other block restrictions correspond to omissions of certain denominator factors.

4. Explicit Evaluations and Special Cases

Explicit formulas for $t^*$ -star values follow from the generating function machinery. Notably:

For pure two-blocks:

$t^*(\{2\}^a) = \beta(2a+1) = \frac{E_{2a}}{2^{2a}(2a)!}$

where $E_{2a}$ are Euler numbers and $\beta(s) = \sum_{k \geq 0} \frac{(-1)^k}{(2k+1)^s}$ is the Dirichlet beta function.

For sandwiched blocks:

$t^*(\{2\}^a, 3, \{2\}^b) = \beta(2a+2b+4) - \beta(2a+2, 2b+2)$

$t^*(\{2\}^a,3,\{2\}^b) = \sum_{r=1}^{a+b+1}(2^{-2r} - 2^{-2(a+b+1-r)}) \beta(2r+1)\beta(2a+2b+3-2r)$

For single "1"-block:

$t^*(\{2\}^a,1,\{2\}^b) = \beta(2a+2b+2) - \beta(2a+1,2b+1)$

(Li et al., 2022, Li et al., 2022, Li et al., 2022).

Connections with alternating $t$ -values allow expressing certain star patterns as weighted alternating sums.

5. Weighted Sum Formulas, Symmetric Sums, and Sum Relations

Weighted sum formulas characterize the global sums of $t$ -harmonic star values, generalizing the classical multiple zeta-sum results:

For any symmetric polynomial $f$ and fixed depth $n$ and weight $k$ with $k\ge n$ ,

$T_f^*(2, k, n) = \sum_{k_1 + \ldots + k_n = k,\,k_i \geq 1} f(k_1, \ldots, k_n) t^*(2k_1, \ldots, 2k_n)$

admits the reduction

$T_f^*(2, k, n) = \sum_{l=0}^{T} c_{f, l}^*(k) (2l)! \, t(2k - 2l)$

where $T = \max\{\lfloor (r+n-2)/2 \rfloor, \lfloor (n-1)/2 \rfloor\}$ for $f$ of degree $r$ and $c_{f, l}^*(k)$ is a degree-bounded symmetric polynomial (Li et al., 2019).

Symmetric sum formulas (Hoffman-type) tie the sum over permutations to set-partitions with explicit combinatorial coefficients: $\sum_{\sigma \in S_n} t^*(k_{\sigma(1)}, \ldots, k_{\sigma(n)}) = \sum_{\pi \in \mathcal{P}_n} c(\pi) \prod_{B \in \pi} t\left(\sum_{i \in B} k_i\right)$ enabling all-star sums with multiple $t$ -values to be reduced to products of single-valued $t$ functions (Li et al., 2019, Li et al., 2022).

Weighted sum formulas for general levels and their block decompositions are available via binomial and exponential generating function identities (Li et al., 2022, Chung, 2016).

6. Algebraic Structure and Relations

Multiple $t$ -harmonic star sums possess analogues of classical MZV algebraic relations:

Stuffle (harmonic) relations: Star product is controlled by $t$ -harmonic stuffle algebras, deforming the ordinary one at $t=0$ and $t=1$ (Murahara et al., 2019).
Duality: $t^*(k_1,\ldots,k_r)$ is $(-1)^r t^*(k_r,\ldots,k_1)$ .
Shuffle relations: These are implemented in suitable algebraic models.
Cyclic and weighted sum relations: Bowman–Bradley and Seki-type identities hold for $t$ -harmonic star values, bridging combinatorics and analytic structure.
Ohno–Zagier relations: The generating functions of $t$ -star values with fixed weight, depth, and height are expressed in terms of very-well-poised ${}_3F_2$ hypergeometric series, generalizing the original Ohno–Zagier result for MZVs (Li et al., 2022, Li et al., 2022).

7. Applications and Recent Developments

Recent advances include

Closed-form evaluations of Apéry-like series: Apéry-type binomial sums involving $t$ -harmonic star sums can be expressed as finite alternating sums of products of Dirichlet beta values:

$\sum_{n=1}^{\infty} \frac{4^n}{n^2 \binom{2n}{n} t_n^*(\{2\}_j)} = 8 \sum_{k=0}^{2j} (-1)^k \beta(k+1) \beta(2j - k + 1)$

for even weights (Layja, 27 Jan 2026).

Evaluations in terms of classical numbers: Secant/tangent connections, via the Euler numbers, provide explicit forms for $t^*(\{2\}^a)$ and related block sums (Li et al., 2022, Chung, 2016).
Level- $N$ and generalized $t$ -star values: Generalization to level- $N$ and development of symmetric and weighted sum frameworks have led to explicit formulas for restricted index sets, including those with indices in $\{1,2,3\}$ and higher (Li et al., 2022).

These results generalize classical zeta phenomenon to the "odd" world of $t$ -values, yielding new identities for sums with intricate block patterns, opening directions for further exploration in the context of modular-type and $q$ -analogue multiple $L$ -values.