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Generalized Euler decomposition formula for interpolated multiple zeta values

Pitu Sarkar, Nita Tamang

TL;DR

The paper extends the Euler decomposition framework to interpolated multiple zeta values by developing a general $t$-shuffle product formula in the noncommutative algebra setting, with a dedicated algebra homomorphism $Z^t$ linking words to $\zeta^t$. It provides two complementary approaches to height-one IMZV decompositions: a combinatorial shuffle analysis and a recursive $t$-shuffle construction, yielding explicit height-one formulas and their equivalence. The work then demonstrates practical applications, deriving alternating-sum identities and relations between MZVs and MZV-star values that interpolate known results and reveal new connections at even weights via duality. Overall, the results generalize classical Euler decompositions and deepen the algebraic understanding of IMZVs, with potential implications for number theory and mathematical physics. $Z^t$-driven Euler-type identities offer a versatile tool for relating IMZVs across heights and depths.

Abstract

In this paper, we obtain a general t-shuffle product formula, using which we derive a generalized Euler decomposition formula for interpolated multiple zeta values. We also provide the same formula in case of height one through two different approaches: one by combinatorial description and another one by recursive formula.

Generalized Euler decomposition formula for interpolated multiple zeta values

TL;DR

The paper extends the Euler decomposition framework to interpolated multiple zeta values by developing a general -shuffle product formula in the noncommutative algebra setting, with a dedicated algebra homomorphism linking words to . It provides two complementary approaches to height-one IMZV decompositions: a combinatorial shuffle analysis and a recursive -shuffle construction, yielding explicit height-one formulas and their equivalence. The work then demonstrates practical applications, deriving alternating-sum identities and relations between MZVs and MZV-star values that interpolate known results and reveal new connections at even weights via duality. Overall, the results generalize classical Euler decompositions and deepen the algebraic understanding of IMZVs, with potential implications for number theory and mathematical physics. -driven Euler-type identities offer a versatile tool for relating IMZVs across heights and depths.

Abstract

In this paper, we obtain a general t-shuffle product formula, using which we derive a generalized Euler decomposition formula for interpolated multiple zeta values. We also provide the same formula in case of height one through two different approaches: one by combinatorial description and another one by recursive formula.
Paper Structure (6 sections, 12 theorems, 112 equations)

This paper contains 6 sections, 12 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['T101']}
  3. Generalized Euler decomposition formula for IMZV of height one
  4. Recursive formula for $t$-shuffle product
  5. Some Applications
  6. Some observations

Key Result

Theorem 1

For integers ${m,u \geq 2, p \geq 1, n,v \geq 0}$, we have where Here for positive integers $l_1, \dots , l_q (,l_{q+1})$ and $n_1, \dots , n_q (,n_{q+1})$ are given by and with $L_j=l_1+l_2+ \dots +l_j,$$N_j= n_1+ \dots + n_j$ for $j \geq 0$ and $L_0=N_0=0$, and $\delta$ is Kronecker's delta symbol given by

Theorems & Definitions (20)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Lemma 2
  • ...and 10 more