Weighted sum formulas of multiple t-values with even arguments

Ce Xu; Zhonghua Li

REVIEW 1 major objections 5 minor 18 references

Weighted sum formulas of multiple t-values with even arguments

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For any symmetric polynomial weight, finite sums of multiple t-values at even arguments equal a short combination of ζ(2l) and t(2k−2l).

desk verdict A genuine, if specialized, extension of weighted sum formulas to multiple t-values; the main theorem holds up and the flagged issues are presentation-level gaps, not mathematical flaws. read the letter →

arxiv 1908.03200 v1 pith:L5DO4V7G submitted 2019-08-05 math.NT

Zhonghua Li , Ce Xu This is my paper · ORCID

classification math.NT MSC 11M3211B68

keywords multiplet-valuest-starvaluesweightedsumformulaevenargumentsBernoullinumberszetasymmetricpolynomials

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that weighted sums of multiple t-values, and their star variants, at even arguments are governed by a single finite shape. For any symmetric polynomial $f\in\mathbb{Q}[x_1,\dots,x_n]$ of degree $r$ and any $k\ge n$, the weighted sum $T_f(2,k,n)$ (and similarly $T_f^\star(2,k,n)$) is written as $\sum_{l=0}^{\min\{T,k\}} c_{f,l}(k)\zeta(2l)t(2k-2l)$, with $T=\max\{[ (r+n-2)/2],[(n-1)/2]\}$ and $c_{f,l}$ a rational-coefficient polynomial in $k$ of controlled degree. Thus no special constants beyond values of $\zeta$ at even integers and one $t$-value are needed. This is the multiple-$t$-value analogue of the conjectured and now proved weighted sum formula for multiple zeta values with even arguments, and it makes these sums effectively computable.

What carries the argument

The central object is the even generating function $F(x)=x/2-x/(e^x+1)=\sum_{i\ge0}\beta_{2i}x^{2i}/(2i)!$, where $\beta_{2i}=(2^{2i}-1)B_{2i}$; it packages the Bernoulli numbers that Euler's formula attaches to $t(2k)$. The operator $D=x\,d/dx$ produces the expansion $D^mF(x)=\sum_{i=0}^{m+1}F_{mi}(x)H(x)^i$ with $H(x)=x/(e^x+1)$, and the triangular matrix $A_m(x)$ built from $F_{mi}$ inverts to express each $H(x)^i$ as a combination of $1,F,DF,\dots,D^mF$. Comparing the even power series on both sides, and using $\mathbb{Q}(x)$-linear independence of $1,F,DF,\dots,D^mF$, forces the leftover coefficient functions $R_j(x)$ to be even, which is what restricts the final answer to the $\zeta(2l)t(2k-2l)$ shape.

What would settle it

Compute the depth-two identity $\sum_{k_1+k_2=k,\,k_1,k_2\ge1} t(2k_1)t(2k_2)=\tfrac12(2k-1)t(2k)$ for $k=10$ by summing over odd $m_1>m_2>0$ up to a large cutoff on both sides; if the two sides differ beyond the numerical tail error, Theorem 2.9 is false and Theorem 1.1 falls with it.

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Extended reading notes

Core claim

On the paper's own terms: Theorem 1.1 states exactly this reduction for both multiple t-values and multiple t-star values, with the same cutoff $T$ and the degree bound $\deg c_{f,l}\le r+n-2l-1$. The route is Theorem 1.2, which drops the symmetry assumption and proves the same finite shape for arbitrary polynomial-weighted products $f(k_1,\dots,k_n)t(2k_1)\cdots t(2k_n)$ over all compositions $k_1+\cdots+k_n=k$. The argument converts each such sum, through Euler's formula for $t(2k)$, into a weighted sum of Bernoulli numbers, evaluates that sum by expanding the generating function under the operator $D=x\,d/dx$, and then converts back via $\zeta(2l)$ and a remaining $t(2k-2l)$. The paper thereby extends the even-argument weighted-sum structure from multiple zeta values to the odd-index variant known as multiple t-values.

Load-bearing premise

The load-bearing premise is the cited-but-not-proved assertion that $1,F(x),DF(x),\dots,D^mF(x)$ are linearly independent over the rational function field $\mathbb{Q}(x)$ for $F(x)=x/2-x/(e^x+1)$; if that independence failed, the proof that each $R_j(x)$ is even would collapse, and with it the restriction of the final formula to the $\zeta(2l)t(2k-2l)$ shape.

Editorial extensions

If this is right

Every symmetric weighted multiple t-value sum and t-star sum with even arguments is reduced to a finite combination of $\zeta(2l)$ and one $t(2k-2l)$; no other special constants appear.
The formulas are effective: the recursive matrix inversion gives a concrete procedure that produces explicit polynomial coefficients for any fixed $f,n,k$, as the appendix demonstrates through depth $4$.
The number of zeta/t terms is bounded by $T=\max\{[ (r+n-2)/2],[(n-1)/2]\}$, which is independent of $k$; increasing the weight only raises the degree of the polynomial coefficients, not the length of the formula.
Theorem 1.2 applies to non-symmetric polynomial weights, so the proof also evaluates arbitrary polynomial-weighted products of individual $t(2k_i)$ values, a statement stronger than the symmetric main theorem.
Specializing $f=1$ yields the unweighted sums $T(2,k,n)$ and $T^\star(2,k,n)$ as particular cases of the same formula.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same generating-function plus matrix-inversion mechanism should transfer to other Bernoulli-type special values whose generating series is even and satisfies a polynomial differential equation; that transfer is not claimed in the paper.
Making the recursion for the inverse matrix $A_m(x)^{-1}$ fully explicit would turn the existence theorem into a closed formula for every coefficient $c_{f,l}(k)$; the paper leaves the coefficients implicit.
The proof's only non-computational input is the linear-independence lemma it cites; a self-contained proof of that lemma for this particular $F$ would make the whole derivation parameter-free.
The evenness of $F$ is essential, so the analogous assertion for odd arguments would likely need a genuinely different mechanism; a natural test is whether any odd-argument analogue holds at all.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a solid, genuine extension of the Li-Qin weighted sum formulas from multiple zeta values to Hoffman's multiple t-values, and the main theorem holds up. The t-case is not a corollary of the zeta case: the generating function F(x)=x/2 - x/(e^x+1) changes the Bernoulli coefficients to beta_{2i}=(2^{2i}-1)B_{2i}, so the coefficient polynomials are different. The paper's contribution is Theorem 1.1 (and the star-value analogue): for any symmetric polynomial f of degree r, the weighted sums T_f(2,k,n) and T*_f(2,k,n) are finite linear combinations of zeta(2l)t(2k-2l) with polynomial coefficients in k of degree at most r+n-2l-1. That is the right shape, matching the zeta-value program.

What the paper does well: the proof is coherent and largely self-contained. The matrix identity (2.5)-(2.8) is a clean way to express powers of H(x) in terms of D^mF(x), and Proposition 2.7's evenness argument is the key step. The appendix examples through depth 4 are extensive and internally consistent; they are useful for checking and would help any reader verify the formulas. The use of Hoffman's symmetric sum formulas (2.17)-(2.18) and the reduction from products of t(2k_i) to weighted sums is natural.

Soft spots: they are presentation-level, not mathematical. First, Proposition 2.7(1) cites [11, Lemma 2.6] for the linear independence of 1,F,DF,...,D^mF over Q(x). That citation is likely fine—since F(x)=(x/2)tanh(x/2) has infinitely many poles, it is not D-finite, so the functions are independent—but the paper should either prove it in one sentence or state the lemma explicitly. As written, the load-bearing evenness of the remainders R_j rests on a pointer, and a referee would want that closed. Second, the proof of Theorem 1.2 is compressed to a single line after Theorem 2.9: it is the linearity from monomial weights to arbitrary polynomial f, but the degree-bound bookkeeping is not written out. Again minor. The reliance on Lemma 2.10 from [11] is fine; that is a published parameter-free lemma.

Who this is for: researchers working on multiple zeta and t-values, weighted sum formulas, and Hoffman's odd variant. It is not a breakthrough, but it is a natural completion of a program and the first weighted polynomial result for t-values. I would cite it. It deserves a serious referee; I would send it out. Recommendation: accept with minor revision after the two presentation gaps are fixed.

Referee Report

1 major / 5 minor

Summary. The paper studies weighted sums of multiple t-values and multiple t-star values at even arguments. For any symmetric polynomial f of degree r in n variables and k≥n, Theorem 1.1 asserts that T_f(2,k,n) and T*_f(2,k,n) can be written as finite sums over l of c_{f,l}(k) ζ(2l) t(2k-2l), with l up to min{T,k}, T=max{[(r+n-2)/2],[(n-1)/2]}, and c_{f,l}∈Q[x] of degree at most r+n-2l-1. The proof develops a Bernoulli-number weighted sum formula (Theorem 2.8) via generating functions and matrix inversion, converts it to t-values using Euler's formula (Theorem 2.9), and reduces the symmetric-weight case to the monomial case using Hoffman's symmetric sum formulas and a polynomial interpolation lemma. The appendix gives explicit formulas through depth 4.

Significance. If correct, the result is a natural and nontrivial analogue for multiple t-values of the weighted sum formulas previously established for multiple zeta values in [11]. The theorem covers all symmetric polynomial weights and gives explicit degree bounds for the coefficient polynomials, and the appendix provides concrete formulas that can be checked numerically. The method is parameter-free and the derivations are largely self-contained apart from cited ingredients. The paper should be of interest to specialists in multiple zeta values and their variants.

major comments (1)

[Section 2.1, after Eq. (2.8)] The proof of Proposition 2.7(1) uses the assertion that 1, F(x), DF(x), ..., D^mF(x) are linearly independent over Q(x), citing [11, Lemma 2.6]. However, the function F(x)=x/2 - x/(e^x+1) that appears here is not the same as the multiple-zeta generating function treated in [11], and the paper does not state the lemma or demonstrate that it applies to this F. This is load-bearing: the independence is exactly what forces each remainder R_j(x) in Proposition 2.7(1) to be even, and evenness is what produces the x^{2l} terms in (2.12) and hence the ζ(2l)t(2k-2l) shape of Theorems 2.8, 2.9, and 1.1. I recommend that the authors state the lemma explicitly and either prove it for this F (for instance, by using the pole structure of F at odd multiples of πi) or quote a version from [11] that is visibly applicable.

minor comments (5)

[Section 1, definition of I(k,n)] The set I(k,n) is defined as all indices of weight k and depth n, but it should be stated explicitly that these are ordered compositions of k into n positive integers; this is essential because the multiple t-values are not symmetric in their arguments.
[Section 2.3, proof of Theorem 1.2] The proof of Theorem 1.2 is reduced to a one-line pointer to Theorem 2.9; since Theorem 2.9 handles monomial weights and both sides of (1.3) are linear in the weight, the step is valid, but the sentence should say 'by linearity' to make the argument transparent.
[Section 2.4, Eq. (2.21) and following] The notation g_{s1,...,si}(x1,...,xi) is confusing: the polynomial should depend on the partition block sizes l_1,...,l_i (and on f), not on the summation variables s_1,...,s_i; a notation such as g_{l_1,...,l_i}(x_1,...,x_i) would be clearer.
[Section 2.2, Proposition 2.7(3)] In the proof of part (3), the text reads 'deg R_j(x) ≤ |m| − j'; this should be '|m|_n − j' for consistency with the definition of |m|_n.
[Throughout] There are a number of OCR-type typographical issues (for example, 'argu ments' in the abstract and 'Masan obu' in the acknowledgments) that should be corrected in the final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the weighted t-value formulas are derived from independent generating-function, Euler, and symmetric-sum ingredients rather than from their own conclusion.

full rationale

The paper's central results (Theorems 1.1, 1.2, 2.8, and 2.9) are obtained by comparing the two series expressions (2.11) and (2.13) for the coefficient of x^{2k} in D^{m_1}F(x)...D^{m_n}F(x); no fitted parameter or pre-assumed target formula enters. Equation (2.15) is Euler's formula, and the passage from Bernoulli weight sums to t-weight sums is a change of basis in powers of pi, not a definitional recycling of the claimed zeta(2l)t(2k-2l) shape. The move from Theorem 2.9 to Theorem 1.2 is linearity in f, and Theorem 1.1 is obtained from Hoffman's symmetric sum formulas (2.17)-(2.18) together with [11, Lemma 4.2], an external parameter-free power-sum lemma. The only substantive citation to the first author's earlier work is the linear independence of 1, F(x), DF(x), ..., D^mF(x) over Q(x) used in Proposition 2.7 and referred to [11, Lemma 2.6]; the manuscript does not prove it for this F, which is a verification gap or correctness risk, but not circularity, because the cited lemma is an independent mathematical statement that does not assume the present formulas and is not fitted to them. No prediction is an input renamed, and no uniqueness or ansatz is imported from the authors' prior papers as a substitute for proof.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The proof relies on standard Euler evaluations, Hoffman's symmetric sum formulas for the t-variant, a polynomial interpolation lemma from the first author's prior zeta-value paper, and a linear independence statement for the generating function. No free parameters are fitted and no new entities are postulated.

assumptions (4)

standard math Euler's evaluation ζ(2k)=(-1)^{k+1}B_{2k}(2π)^{2k}/(2(2k)!), and t(2k) expressed in terms of β_{2k}.
Used at (2.15) to convert Bernoulli-number weighted sums into t-value weighted sums; classical result.
domain assumption Hoffman's symmetric sum formulas (2.17) and (2.18) for multiple t-values and t-star values.
Cited from [7, Theorems 2.5 and 2.8]; the proof of Theorem 1.1 reduces ordinary multiple t-values to products of single t-values through these formulas.
domain assumption Lemma 2.10 from [11, Lemma 4.2]: sums of monomials over compositions equal a polynomial in the total sum.
Used in Section 2.4 to convert sums over k with fixed block sums into polynomial weights for Theorem 1.2.
ad hoc to paper Linear independence of 1,F,DF,...,D^mF over Q(x) for F(x)=x/2 - x/(e^x+1).
Invoked in Proposition 2.7 to conclude each R_j is even; the paper refers to [11, Lemma 2.6] but no proof is given for this F.

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Cite this review

Pith. "Pith review of Weighted sum formulas of multiple t-values with even arguments." pith.science (2026). https://pith.science/paper/L5DO4V7G

@misc{pith2026190803200,
  author       = {Pith},
  title        = {Pith review of: Weighted sum formulas of multiple t-values with even arguments},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L5DO4V7G}},
  note         = {Machine review of arXiv:1908.03200}
}

read the original abstract

In this paper, we study the weighted sums of multiple t-values and of multiple t-star values at even arguments. Some general weighted sum formulas are given, where the weight coefficients are given by (symmetric) polynomials of the arguments.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 17 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [11]

Li and C

Z. Li and C. Qin, Weighted sum formulas of multiple zeta values with even arguments, Math. Z., 2019, 291, 1337-1356

work page 2019

[2] [1]

Eie, W-C

M. Eie, W-C. Liaw and Y.L. Ong, Several weighted sum formula of mu ltiple zeta values, Int. J. Number Theory, 2017, 13(9), 2253-2264

work page 2017

[3] [2]

Eie and Y.L

M. Eie and Y.L. Ong, Sum formulas of multiple zeta values with even ar guments and polynomial weights, J. Number Theory, 2018, 188, 247-262

work page 2018

[4] [3]

Gangl, M

H. Gangl, M. Kaneko and D. Zagier, Double zeta values and modular forms, in: Proceedings of the Conference in Memory of Tsuneo Arakawa, 2006, 71-106

work page 2006

[5] [4]

Genˇ cev, On restricted sum formulas for multiple zeta values w ith even arguments, Arch

M. Genˇ cev, On restricted sum formulas for multiple zeta values w ith even arguments, Arch. Math., 2016, 107, 9-22

work page 2016

[6] [5]

L. Guo, P. Lei and J. Zhao, Families of weighted sum formulas for m ultiple zeta values, Int. J. Number Theory, 2015, 11(3), 997-1025

work page 2015

[7] [6]

Hoﬀman, Multiple harmonic series, Paciﬁc J

M.E. Hoﬀman, Multiple harmonic series, Paciﬁc J. Math., 1992, 152(2), 275-290

work page 1992

[8] [7]

Hoﬀman, An odd variant of multiple zeta values, Comm

M.E. Hoﬀman, An odd variant of multiple zeta values, Comm. Number Theory Phys., to appear, arXiv:1612.05232v4 [math.NT]

arXiv