On a conjecture of Kaneko and Ohno

TL;DR

This paper proves Kaneko and Ohno's conjecture that certain differences of sums of multiple zeta-star values can be expressed as polynomials of zeta values with rational coefficients.

Contribution

The paper provides a proof confirming that the difference of specific multiple zeta-star value sums can be represented as rational-coefficient polynomials of zeta values.

Findings

Confirmed the conjecture for all positive integers m, n, s with m,n ≥ s.

Established the polynomial expression of the difference in terms of zeta values.

Contributed to the understanding of the algebraic structure of multiple zeta-star values.

Abstract

Let $X_0^{\star}(k,n,s)$ denote the sum of all multiple zeta-star values of weight $k$, depth $n$ and height $s$. Kaneko and Ohno conjecture that for any positive integers $m,n,s$ with $m,n\geqslant s$, the difference $(-1)^mX_0^{\star}(m+n+1,n+1,s)-(-1)^nX_0^{\star}(m+n+1,m+1,s)$ can be expressed as a polynomial of zeta values with rational coefficients. We give a proof of this conjecture in this paper.