Sum formula for finite multiple zeta values

Abstract

The sum formula is one of the most well-known relations among multiple zeta values. This paper proves a conjecture of Kaneko predicting that an analogous formula holds for finite multiple zeta values.

Ohno-type identities for multiple harmonic sums

“…Let k = ({1} i−1 , 2, {1} r−i ) and e = k − r − 1 in (17). Then k ∨ = (i, r − i + 1) and we have (14) and Theorem 2.4. Since k is even, one of B p−k+r+j or B p−j−r−1 is zero.…”

“…from the identity (6) in the samy way as (18). Let us fix 0 ≤ j < k − r. By (14) and Proposition 4.6, if j + r is odd, then ζ ⋆ A 3 ({1} k−r−j ) is divisible by p and T ⋆ j+r,r is divisible by p 2 and if j + r is even, then ζ ⋆ A 3 ({1} k−r−j ) is divisible by p 2 and T ⋆ j+r,r is divisible by p. Therefore, ζ ⋆ A 3 ({1} k−r−j )T ⋆ j+r,r = 0 and we see that the left hand side of (23) is equal to T ⋆ k,r . On the other hand, by using Proposition 4.1, Proposition 4.2 and (22), we see that the right hand side of (23) is equal to Let us fix 1 ≤ j ≤ r − 1 and j ≤ l ≤ k − r + j.…”

A Note on Finite Real Multiple Zeta Values

“…Thus, S k,n,i enjoy the recursion relation in the following lemma, which can be proved in the same way as in [11…”

Cyclotomic analogues of finite multiple zeta values

“…They conjecture that the finite multiple zeta values satisfy the same Qlinear relation as the symmetric multiple zeta values and vice versa (see Conjecture 3.10). A few families of Q-linear relations which are satisfied by the finite and the symmetric multiple zeta values simultaneously are obtained by the works of Murahara, Saito and Wakabayashi in [11,15], where the star versions ζ ⋆ A (k) and ζ ⋆ S (k) are also considered. In the present paper, we examine for n ∈ Z 1 the values z n (k; ζ n ) and z ⋆ n (k; ζ n ) of finite multiple harmonic q-series z n (k; q) and z ⋆ n (k; q) evaluated at a primitive n-th root of unity ζ n (see Definition 2.1).…”