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Symmetry results for multiple $t$-values

Steven Charlton, Michael E. Hoffman

TL;DR

This work establishes a regularized symmetry for multiple $t$-values under index reversal: for a composition $I$ of length $n\ge3$, $\mathrm{reg}^*_T t(I)=(-1)^{n-1}\mathrm{reg}^*_T t(\overline{I})\pmod{\text{products}}$, with regularization carried out at $T=\log 2$ and extended to interpolated $t$-values. The authors develop a comprehensive framework built on regularized polylogarithms, asymptotic expansions, and generating-series methods, enabling a truncated-to-full-limit passage and yielding a general symmetry identity that also yields product-term corrections via Murakami-type relations between $\zeta$ and $t$. They then apply this machinery to derive explicit evaluations for whole families of MtVs, including $t(3,\{2\}^n,3)$, $t(1,\{2\}^n,1)$, and $t^{1/2}(\{1\}^n,2\ell+2)$, expressing results in terms of $\zeta$ values, Dirichlet beta values, and powers of $\log 2$, and revealing rich depth-structure and parity phenomena. The results connect to Ohno–Zagier-type summations and Murakami’s basis results for motivic MtVs, highlighting deep algebraic relations within the MtV algebra and offering practical closed-form evaluations for a broad class of values. Overall, the paper advances the understanding of symmetry and regularization in multiple (alternating) zeta/t-values and provides explicit, computable formulas across several important families of indices.

Abstract

For a composition $I$ whose first part exceeds 1, we can define the multiple $t$-value $t(I)$ as the sum of all the terms in the series for the multiple zeta value $ζ(I)$ whose denominators are odd. In this paper we show that if $I$ is composition of $n\ge 3$, then $t(I)=(-1)^{n-1}t(\bar I)$ mod products, where $\bar I$ is the reverse of $I$, and both sides are suitably regularized when $I$ ends in 1. This result is not true for multiple zeta values, though there is an argument-reversal result that does hold for them (and for multiple $t$-values as well). We actually prove a more general version of this result, and then use it to establish explicit formulas for several classes of multiple $t$-values and interpolated multiple $t$-values.

Symmetry results for multiple $t$-values

TL;DR

This work establishes a regularized symmetry for multiple -values under index reversal: for a composition of length , , with regularization carried out at and extended to interpolated -values. The authors develop a comprehensive framework built on regularized polylogarithms, asymptotic expansions, and generating-series methods, enabling a truncated-to-full-limit passage and yielding a general symmetry identity that also yields product-term corrections via Murakami-type relations between and . They then apply this machinery to derive explicit evaluations for whole families of MtVs, including , , and , expressing results in terms of values, Dirichlet beta values, and powers of , and revealing rich depth-structure and parity phenomena. The results connect to Ohno–Zagier-type summations and Murakami’s basis results for motivic MtVs, highlighting deep algebraic relations within the MtV algebra and offering practical closed-form evaluations for a broad class of values. Overall, the paper advances the understanding of symmetry and regularization in multiple (alternating) zeta/t-values and provides explicit, computable formulas across several important families of indices.

Abstract

For a composition whose first part exceeds 1, we can define the multiple -value as the sum of all the terms in the series for the multiple zeta value whose denominators are odd. In this paper we show that if is composition of , then mod products, where is the reverse of , and both sides are suitably regularized when ends in 1. This result is not true for multiple zeta values, though there is an argument-reversal result that does hold for them (and for multiple -values as well). We actually prove a more general version of this result, and then use it to establish explicit formulas for several classes of multiple -values and interpolated multiple -values.
Paper Structure (38 sections, 26 theorems, 269 equations)

This paper contains 38 sections, 26 theorems, 269 equations.

Table of Contents

  1. Introduction
  2. Outline.
  3. Acknowledgements.
  4. Regularized version of the Symmetry Theorem
  5. Polylogarithms and regularization
  6. Asymptotic expansions of polylogarithms
  7. Asymptotic expansions for zeta and t values
  8. Truncated identity
  9. Evaluation 1:
  10. Evaluation 2:
  11. Regularization statement
  12. Case 1, $\ \uline{{\boldsymbol \phi}} \neq \uline{{\boldsymbol 0}}$ and $\phi_1 + \cdots + \phi_m \neq 0$:
  13. Case 2, $\uline{{\boldsymbol \phi}} \neq \uline{{\boldsymbol 0}}$ and $\phi_1 + \cdots + \phi_m = 0$:
  14. Case 3: $\uline{{\boldsymbol \phi}}=\uline{{\boldsymbol 0}}$.
  15. Case $m = 1$:
  16. ...and 23 more sections

Key Result

Theorem 1.1

If $I$ is a composition of $n\ge 3$, then where $\bar{I}$ is the reverse of $I$.

Theorems & Definitions (62)

  • Theorem 1.1: Symmetry Theorem
  • Corollary 1.2
  • proof
  • Theorem 1.3: Stuffle antipode
  • proof
  • Corollary 1.4
  • Remark 1.5
  • Definition 2.1: Multiple polylogarithm
  • Definition 2.2: burgos-fresan
  • Example 2.3
  • ...and 52 more