Multiple Zeta Star Functions

• Multiple zeta star functions are star variants of Euler–Zagier multiple zeta functions, defined using weak inequalities that enable studies in analytic, combinatorial, and arithmetic frameworks.
• They utilize integral representations, generating functions, and determinant identities to derive explicit special value formulas and connect to Schur-type generalizations.
• Recent research employs Stirling and Bell polynomials to evaluate MZSFs at non-positive integers, revealing deep links with regular multiple zeta functions and positive-characteristic theories.

Multiple zeta star functions (MZSFs) are star-variants of Euler–Zagier multiple zeta functions in which weak inequalities replace strict ones. In depth $r$, they are given by

$$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$

and the corresponding multiple zeta-star values arise by specializing to positive integer indices. Recent work places MZSFs at the intersection of analytic continuation, iterated integrals, generating functions, Schur-theoretic determinant identities, Bell- and Stirling-polynomial expansions, positive-characteristic zeta theory, and Diophantine approximation. The subject is now broad enough that the same object is routinely studied as a multivariable meromorphic function, a period-like integral, a quasi-symmetric specialization, and a source of explicit special-value formulae (Ishii et al., 18 Aug 2025).

1. Definitions, domains, and basic variants

For an index $\mathbf{k}=(k_1,\ldots,k_n)$ of positive integers with $k_1\ge 2$, the multiple zeta-star value is

$$\zeta^\star(\mathbf{k})=\sum_{m_1\ge \cdots \ge m_n\ge 1}\frac{1}{m_1^{k_1}\cdots m_n^{k_n}}.$$

The weak inequalities distinguish the star theory from the ordinary multiple zeta function, where the summation is strict. In the function-theoretic setting, $\zeta_r^\star(s_1,\dots,s_r)$ converges in a domain $D_r$, admits meromorphic continuation, and has known singularities. For identical arguments, the standard shorthand is

$\zeta_r^\star(s):=\zeta_r^\star(s,\ldots,s),$

with the convention $\zeta_0^\star(\varnothing)=1$ (Ishii et al., 18 Aug 2025, Mehta, 28 Mar 2026).

At non-positive integers, two limiting procedures are used. For $l=(l_1,\dots,l_r)\in \mathbb Z_{\ge 0}^r$,

$$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$0

while

$$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$1

These regular and reverse values play a central role in explicit evaluation theorems at non-positive integers (Ishii et al., 18 Aug 2025).

The same star principle extends to a substantial family of variants. Finite and symmetric multiple zeta-star values support Ohno-type generating functions (Hirose et al., 2019). Multiple Hurwitz zeta-star functions admit analogous sum formulae (Chen, 2017). Multiple star $$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$2-values

$$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$3

form the odd-denominator analogue of MZSFs (Chung, 2016). In positive characteristic, one has $$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$4-adic and $$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$5-adic multiple zeta star functions defined by weakly decreasing degree indices rather than integer summation variables (Matsuzuki, 2022).

2. Relations with ordinary multiple zeta functions and Schur-type generalizations

A foundational fact is that MZSFs and ordinary multiple zeta functions (MZFs) are mutually convertible by summing over all ways of merging adjacent arguments: $$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$6 where $$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$7 is the length of $$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$8. This already shows that the star operation is not merely a notational variant: it is tied to a specific composition-refinement combinatorics (Nakasuji et al., 2017).

That relation becomes especially sharp at non-positive integers. If $$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$9 with $\mathbf{k}=(k_1,\ldots,k_n)$0, then

$\mathbf{k}=(k_1,\ldots,k_n)$1

where $\mathbf{k}=(k_1,\ldots,k_n)$2. A common misconception is that star and non-star values are always substantially different; the non-positive integer theory shows that, under this hypothesis, the difference collapses to a sign, even though other parts of the theory retain distinct combinatorial content (Ishii et al., 18 Aug 2025).

Schur multiple zeta functions provide a combinatorial interpolation between MZFs and MZSFs. For a partition $\mathbf{k}=(k_1,\ldots,k_n)$3, the Schur multiple zeta function is a sum over semi-standard Young tableaux. The one-column case $\mathbf{k}=(k_1,\ldots,k_n)$4 recovers the ordinary multiple zeta function, while the one-row case $\mathbf{k}=(k_1,\ldots,k_n)$5 recovers the MZSF. Under content parametrization, hook-type Schur multiple zeta functions decompose explicitly into products of ordinary and star zeta functions: $\mathbf{k}=(k_1,\ldots,k_n)$6 and arbitrary shapes are then reconstructed via the Giambelli determinant. This places MZSFs inside a larger Schur-theoretic framework governed by Frobenius coordinates, determinant identities, and root-system analogues (Matsumoto et al., 2023, Nakasuji et al., 2017).

3. Explicit special values: Stirling polynomials, Bell polynomials, and singularities

A major recent development is the explicit evaluation of MZSFs at non-positive integers by Stirling polynomials. Let $\mathbf{k}=(k_1,\ldots,k_n)$7, $\mathbf{k}=(k_1,\ldots,k_n)$8, $\mathbf{k}=(k_1,\ldots,k_n)$9, and $k_1\ge 2$0. If $k_1\ge 2$1 denotes the Stirling polynomial of the second kind, defined by

$k_1\ge 2$2

then the reverse value of the depth-$k_1\ge 2$3 MZSF is given explicitly by

$k_1\ge 2$4

The same paper gives recurrences and a Gregory-coefficient expansion for reverse values, extending a line of work associated with Matsusaka, Murahara, and Onozuka. The generalized Gregory coefficients $k_1\ge 2$5 are defined by

$k_1\ge 2$6

and reverse values are expressed as finite sums built from these coefficients and Stirling-polynomial weights (Ishii et al., 18 Aug 2025).

For identical arguments, harmonic-product recurrences lead to Bell-polynomial formulae. For $k_1\ge 2$7 and $k_1\ge 2$8,

$k_1\ge 2$9

and more explicitly

$$\zeta^\star(\mathbf{k})=\sum_{m_1\ge \cdots \ge m_n\ge 1}\frac{1}{m_1^{k_1}\cdots m_n^{k_n}}.$$0

Equivalently,

$$\zeta^\star(\mathbf{k})=\sum_{m_1\ge \cdots \ge m_n\ge 1}\frac{1}{m_1^{k_1}\cdots m_n^{k_n}}.$$1

where $$\zeta^\star(\mathbf{k})=\sum_{m_1\ge \cdots \ge m_n\ge 1}\frac{1}{m_1^{k_1}\cdots m_n^{k_n}}.$$2 is the complete Bell polynomial. This makes the singularity structure accessible through the poles of $$\zeta^\star(\mathbf{k})=\sum_{m_1\ge \cdots \ge m_n\ge 1}\frac{1}{m_1^{k_1}\cdots m_n^{k_n}}.$$3. The same work notes that the star function has the same possible singularities as the non-star function at the points $$\zeta^\star(\mathbf{k})=\sum_{m_1\ge \cdots \ge m_n\ge 1}\frac{1}{m_1^{k_1}\cdots m_n^{k_n}}.$$4, and that the Bell-polynomial formula gives a direct route to residues and pole orders. It also records special values such as

$$\zeta^\star(\mathbf{k})=\sum_{m_1\ge \cdots \ge m_n\ge 1}\frac{1}{m_1^{k_1}\cdots m_n^{k_n}}.$$5

These formulae complement the non-positive integer theory by showing that the identical-argument specialization has its own closed algebraic model (Mehta, 28 Mar 2026).

4. Generating functions, block structures, and explicit families

Generating functions are one of the most effective ways to organize MZSF families. For repeated blocks of length $$\zeta^\star(\mathbf{k})=\sum_{m_1\ge \cdots \ge m_n\ge 1}\frac{1}{m_1^{k_1}\cdots m_n^{k_n}}.$$6,

$$\zeta^\star(\mathbf{k})=\sum_{m_1\ge \cdots \ge m_n\ge 1}\frac{1}{m_1^{k_1}\cdots m_n^{k_n}}.$$7

For more complicated patterns, generating series can often be converted into hypergeometric expressions and then attacked by creative telescoping (Au et al., 2024).

One influential line of work develops generating functions for arbitrary patterns of blocks of $$\zeta^\star(\mathbf{k})=\sum_{m_1\ge \cdots \ge m_n\ge 1}\frac{1}{m_1^{k_1}\cdots m_n^{k_n}}.$$8s and intervening indices, expressing MZSFs in terms of multiple alternating Euler sums and reducing the lengths of blocks of twos. In particular, explicit series are obtained for

$$\zeta^\star(\mathbf{k})=\sum_{m_1\ge \cdots \ge m_n\ge 1}\frac{1}{m_1^{k_1}\cdots m_n^{k_n}}.$$9

with the coefficients controlled by multiple sharp sums $\zeta_r^\star(s_1,\dots,s_r)$0. This yields a systematic block-reduction principle for star values that had previously been available only in special configurations (Pilehrood et al., 2018).

Specific index families have been worked out in closed form. For the 3–2–1 sector, generating-function methods and Bell-polynomial expansions yield explicit formulae for $\zeta_r^\star(s_1,\dots,s_r)$1, $\zeta_r^\star(s_1,\dots,s_r)$2, and the generalized families

$\zeta_r^\star(s_1,\dots,s_r)$3

including product formulae for the corresponding generating functions in terms of trigonometric factors and coefficient formulae in Bernoulli numbers (Pilehrood et al., 2018). A different strand, based on creative telescoping, establishes new evaluations for block patterns such as $\zeta_r^\star(s_1,\dots,s_r)$4, reducing them to polynomials in Riemann zeta values after first passing through interpolated half-values: $\zeta_r^\star(s_1,\dots,s_r)$5 The paper also treats $\zeta_r^\star(s_1,\dots,s_r)$6, $\zeta_r^\star(s_1,\dots,s_r)$7, and $\zeta_r^\star(s_1,\dots,s_r)$8 by the same pipeline (Au et al., 2024).

Finite and symmetric star values support their own generating-function theory. For an index $\zeta_r^\star(s_1,\dots,s_r)$9, Kaneko’s conjecture for $D_r$0 was confirmed and generalized to arbitrary depth-three indices. If

$D_r$1

then for $D_r$2,

$D_r$3

and for $D_r$4,

$D_r$5

These formulas encode duality through the Hoffman dual and show that, in the finite and symmetric settings, depth-three Ohno-type sums reduce to quadratic forms in depth-one series (Hirose et al., 2019).

5. Integral, cyclic, and combinatorial frameworks

An integral theory for MZSFs parallel to the classical iterated-integral theory of ordinary multiple zeta values was established by Yamamoto. For an index $D_r$6, let $D_r$7 and

$D_r$8

With $D_r$9 and $\zeta_r^\star(s):=\zeta_r^\star(s,\ldots,s),$0, one has

$\zeta_r^\star(s):=\zeta_r^\star(s,\ldots,s),$1

where $\zeta_r^\star(s):=\zeta_r^\star(s,\ldots,s),$2 is determined by alternating inequalities $\zeta_r^\star(s):=\zeta_r^\star(s,\ldots,s),$3 or $\zeta_r^\star(s):=\zeta_r^\star(s,\ldots,s),$4 according to membership in $\zeta_r^\star(s):=\zeta_r^\star(s,\ldots,s),$5. For $\zeta_r^\star(s):=\zeta_r^\star(s,\ldots,s),$6,

$\zeta_r^\star(s):=\zeta_r^\star(s,\ldots,s),$7

This integral formalism is subsumed by the more general theory of admissible 2-labeled posets. If $\zeta_r^\star(s):=\zeta_r^\star(s,\ldots,s),$8 is such a poset, then

$\zeta_r^\star(s):=\zeta_r^\star(s,\ldots,s),$9

with multiplicativity, decomposition under refinement of incomparable elements, and duality $\zeta_0^\star(\varnothing)=1$0. The same framework covers MZVs, MZSFs, Arakawa–Kaneko zeta values, Mordell–Tornheim zeta values, and root-system zeta functions of type $\zeta_0^\star(\varnothing)=1$1 (Yamamoto, 2014).

Cyclic relations extend the algebraic structure beyond integral representations. The cyclic relation of Hirose, Murakami, and Murahara was generalized to complex variables, and the specialization to positive integers recovers the cyclic sum formula for multiple zeta-star values: $\zeta_0^\star(\varnothing)=1$2 The complex-variable version places star and non-star cyclic identities inside a unified analytic framework that also includes derivation relations (Murahara et al., 2020).

Several explicitly evaluable star families were obtained by integral and iterated-integral methods. Examples include polynomial expressions for

$\zeta_0^\star(\varnothing)=1$3

in terms of classical zeta values, polylogarithms, and powers of $\zeta_0^\star(\varnothing)=1$4 (Xu, 2017), as well as explicit relations between MZSVs, Kaneko–Yamamoto type MZVs, alternating MZVs, and the Apéry-type multiple zeta $\zeta_0^\star(\varnothing)=1$5-star values

$\zeta_0^\star(\varnothing)=1$6

which are shown in many cases to be $\zeta_0^\star(\varnothing)=1$7-linear combinations of alternating MZVs (Xu, 2020).

6. Arithmetic extensions, positive characteristic, and limiting-set phenomena

In positive characteristic, MZSFs admit $\zeta_0^\star(\varnothing)=1$8-adic and $\zeta_0^\star(\varnothing)=1$9-adic analogues. With $l=(l_1,\dots,l_r)\in \mathbb Z_{\ge 0}^r$0, $l=(l_1,\dots,l_r)\in \mathbb Z_{\ge 0}^r$1, and $l=(l_1,\dots,l_r)\in \mathbb Z_{\ge 0}^r$2, the $l=(l_1,\dots,l_r)\in \mathbb Z_{\ge 0}^r$3-adic star function is

$l=(l_1,\dots,l_r)\in \mathbb Z_{\ge 0}^r$4

while the $l=(l_1,\dots,l_r)\in \mathbb Z_{\ge 0}^r$5-adic version is

$l=(l_1,\dots,l_r)\in \mathbb Z_{\ge 0}^r$6

Both are rigid analytic; for $l=(l_1,\dots,l_r)\in \mathbb Z_{\ge 0}^r$7 they recover the single-variable zeta functions of Carlitz–Goss and Goss. A central relation is the orthogonal property

$l=(l_1,\dots,l_r)\in \mathbb Z_{\ge 0}^r$8

with a parallel identity in the $l=(l_1,\dots,l_r)\in \mathbb Z_{\ge 0}^r$9-adic setting. The same theory gives integral expressions for negative integer values, Kummer-type congruences such as

$$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$00

and recursive links between $$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$01-adic and $$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$02-adic star values (Matsuzuki, 2022).

A separate arithmetic direction studies the set of multiple zeta-star values itself as a subset of the real line. The set $$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$03 of all multiple zeta-star values is a countable dense subset of $$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$04. A Dirichlet-type criterion characterizes when $$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$05 belongs to $$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$06, and a Khinchin-type zero-one law states that for

$$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$07

one has $$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$08 if $$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$09 converges and $$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$10 if it diverges (Li, 30 Mar 2025).

Finite multiple star harmonic sums admit an analytic interpolation built from divided differences. For fixed $$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$11,

$$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$12

and the interpolated function $$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$13 has the integral representation

$$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$14

where $$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$15 is a divided difference and the Hermite–Genocchi formula is used to prove analyticity. For real $$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$16, the resulting map $$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$17 is a bijection onto $$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$18, giving a finite zeta-star correspondence; for complex $$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$19 with $$\zeta_r^\star(s_1,\dots,s_r)=\sum_{0<n_1\le n_2\le \cdots \le n_r}\frac{1}{n_1^{s_1}\cdots n_r^{s_r}},$$20, injectivity is conjectured (Li, 15 Jun 2026).

Taken together, these developments show that MZSFs are no longer studied only as a star-version of Euler–Zagier sums. They form a nexus joining explicit special-value theory, determinant and tableau formalisms, iterated-integral and generating-function methods, positive-characteristic interpolation, and metric questions about the distribution and coding of zeta-star values themselves.