[Paper Review] Ohno-type relation for finite multiple zeta values

Kojiro Oyama|arXiv (Cornell University)|Jun 2, 2015

Advanced Mathematical Identities5 references4 citations

TL;DR

This paper proves an Ohno-type relation for finite multiple zeta values (FMZVs), confirming a conjecture by Kaneko. Using algebraic structures in the non-commutative polynomial algebra ℋ¹ and properties of Hoffman's dual, the authors establish a duality between sums of FMZVs under index shifts, leading to an alternative proof of the sum formula for FMZVs first proved by Saito and Wakabayashi.

ABSTRACT

Ohno's relation is a well-known relation among multiple zeta values.In this paper, we prove Ohno-type relation for finite multiple zeta values, which is conjectured by Kaneko.As a corollary, we give an alternative proof of the sum formula for finite multiple zeta values, which was first proved by Saito and Wakabayashi.

Motivation & Objective

• To prove an Ohno-type relation for finite multiple zeta values (FMZVs), confirming a conjecture by Kaneko.
• To establish a duality between sums of FMZVs under index shifts, linking them to Hoffman's dual indices.
• To provide an alternative algebraic proof of the sum formula for FMZVs, previously established by Saito and Wakabayashi.
• To extend the framework of Ohno's relation to the finite zeta setting using the algebraic structure of ℋ¹ and the map T.
• To demonstrate that the sum formula for FMZVs follows as a corollary from the main duality relation.

Proposed method

• Define the finite multiple zeta value ζₐ(k) as an element of the algebra 𝒜 = (∏ₚ ℤ/ₚℤ)/(⊕ₚ ℤ/ₚℤ), using p-adic reductions of harmonic sums.
• Use the non-commutative polynomial algebra ℋ¹ = ℚ⟨x,y⟩y, with generators zₖ = xᵏ⁻¹y, to represent FMZVs via the map Zₐ: ℋ¹ → 𝒜.
• Employ the harmonic product * and shuffle product X on ℋ¹, with Zₐ preserving the harmonic product and transforming the shuffle product via sign and reversal.
• Introduce the map T: ℋ¹ → ℋ¹ that implements Hoffman's dual on indices, satisfying Zₐ(T(w)) = ζₐ(k∨) when Zₐ(w) = ζₐ(k).
• Apply the identity involving the generating function 1/(1−yu)*w = 1/(1−yu) X Δᵤ(w) to derive the main duality relation.
• Use inclusion-exclusion on the number of non-zero components in exponent vectors to equate sums over shifted indices, proving the main theorem via combinatorial counting.

Experimental results

Research questions

• RQ1 Does an Ohno-type relation hold for finite multiple zeta values, as conjectured by Kaneko?
• RQ2 Can the sum formula for finite multiple zeta values be derived algebraically from a more general duality relation?
• RQ3 How does Hoffman's duality on indices manifest in the finite zeta setting through the algebraic structure of ℋ¹?
• RQ4 What is the precise relationship between sums of FMZVs under index shifts and their duals under the harmonic product?
• RQ5 Can the generating function approach using Δᵤ and the shuffle product be used to prove identities in the finite zeta setting?

Key findings

• The main theorem establishes a duality: ∑_{|e|=n} ζₐ(k₁+e₁,…,kᵣ+eᵣ) = ∑_{|e′|=n} ζₐ((k′₁+e′₁,…,k′ₛ+e′ₛ)∨), where (k′₁,…,k′ₛ) = (k₁,…,kᵣ)∨.
• The proof confirms Kaneko’s conjecture on the Ohno-type relation for finite multiple zeta values.
• The sum formula for FMZVs is recovered as a corollary, with the right-hand side expressed as (−1)ⁱ⁻¹(ℬₖ₋₁ⁱ⁻¹ + (−1)ʳℬₖ₋₁ʳ⁻ⁱ)(Bₚ₋ₖ/k mod p)ₚ.
• The identity holds for all k,r,i ≥ 1 with 1 ≤ i ≤ r ≤ k−1, and the result is consistent for both even and odd k due to properties of Bernoulli numbers.
• The coefficient of the sum formula is shown to be (−1)ⁱ⁻¹(−(−1)ᵏ⁺¹ℬₖ₋₁ⁱ⁻¹ + (−1)ʳℬₖ₋₁ʳ⁻ⁱ), which matches the known sum formula under p ≥ k+3.
• The proof relies on inclusion-exclusion over non-zero components in exponent vectors, showing each index appears exactly once in the alternating sum.