A proof of a weighted sum conjecture for finite multiple zeta values of level two

Abstract

M. Kaneko, T. Murakami and A. Yoshihara introduced finite multiple zeta values of level two and conjectured a weighted sum formula for indices whose components belong to $\{1,2\}$. In this paper, we use generating functions and linear recurrence relations to prove the conjecture. More precisely, we reduce the problem to a polynomial identity and solve the resulting second-order difference equation by identifying its specialized solutions with ${}_4{F}_3$ hypergeometric polynomials of Racah-type.