Weighted sum formula for multiple harmonic sums modulo primes

Abstract

In this paper we prove a weighted sum formula for multiple harmonic sums modulo primes, thereby proving a weighted sum formula for finite multiple zeta values. Our proof utilizes difference equations for the generating series of multiple harmonic sums. We also conjecture several weighted sum formulas of similar flavor for finite multiple zeta values.

Restricted sum formula for finite and symmetric multiple zeta values

“…For example, we have G 1 ((2, 3), 1) = (3, 3)+ (2,4). For positive integers k, r, i with 1 ≤ i ≤ r ≤ k, we put Theorem 2.3 (Hoffman [3], Jarossay [5]).…”

“…In the following, we use the letter F stands for either A or S, e.g., the symbol ζ F means ζ A or ζ S . Hirose, Saito, and the author [2] proved the following weighted sum formula for FMZ(S)Vs. In this paper, we will give its alternative proof.…”

“…Hirose, Saito, and the author [2] proved the following weighted sum formula for FMZ(S)Vs. In this paper, we will give its alternative proof.…”

“…k, l) := wt(e)=l dep(e)=dep(k) (k ⊕ e) ∈ I,G 2 (k, l) := wt(e)=l dep(e)=dep(k ∨ ) (k ∨ ⊕ e) ∨ ∈ I, G(k, l) := G 1 (k, l) − G 2 (k, l) ∈ I.For example, we have G 1((2, 3), 1) = (3, 3)+(2,4). For positive integers k, r, i with 1 ≤ i ≤ r ≤ k, we put H(k, r, i)…”

Yamamoto's interpolation of finite multiple zeta and zeta-star values

“…In Section 3, we prove the Bowman-Bradley type formulas (Theorems 1.3 and 3.1) by comparing the coefficients of 'Bowman-Bradley sum' in the two successive degrees in t-FMZVs. In Section 4, we partially interpolate the weighted sum formulas obtained by Hirose-Murahara-Saito [7] for A-FMZ(S)Vs and Murahara [27] for FMZ(S)Vs. In Section 5, we introduce several formulas that are relatively easy to obtain, such as the harmonic relation, the shuffle relation, the duality relation, and the derivation relation.…”