On Generalizations of Weighted Sum Formulas of Multiple Zeta Values

Abstract

In this paper, we compute shuffle relations from multiple zeta values of the form ζ({1}m-1, n+1) or sums of multiple zeta values of fixed weight and depth. Some interesting weighted sum formulas are obtained, such as [Formula: see text] where m and k are positive integers with m ≥ 2k. For k = 1, this gives Ohno–Zudilin's weighted sum formula.

Restricted sum formula for finite and symmetric multiple zeta values

“…Remark 1.2. We note that similar weighted sum formulas for MZVs are known (see Guo-Xie [1], Ohno-Zudilin [10], and Ong-Eie-Liaw [11]). Kamano [6] also obtained somewhat different weighted sum formulas for FMZVs. )…”

A parameterized generalization of the sum formula for quadruple zeta values

“…Various generalizations of the sum formula have been studied: Ohno's relations, the cyclic, restricted and weighted sum formulas [1,5,8,10,14,15,16,17,18,19,20]. Recently parameterized generalizations of the sum formula, which we call parameterized sum formulas, were given for double and triple zeta values [3,13].…”

“…In this paper, we give a parameterized sum formula for quadruple zeta values which has four parameters and is invariant under a cyclic group of order four. By substituting special values for the parameters, we also obtain weighted sum formulas for quadruple zeta values which contain results of Guo and Xie [5] and of Ong, Eie and Liaw [19].…”

“…As applications of the parameterized sum formula, we give the weighted sum formulas in Theorem 1.2 below, which consist of the known formulas (1.4), (1.5) and (1.6), and the new formula (1.7) whose weights have not only powers of 2 but also powers of 3. To be exact, (1.4) and (1.6) were the results of [5,Theorem 1.1] and [19,Main Theorem] for quadruple zeta values, respectively. It seems that (1.5) is not printed out anywhere, but (1.5) is easily derived from subtracting (1.6) from twice (1.4).…”

On the convolutions of sums of multiple zeta(-star) values of height one

“…On the simplex D 1 , the Remark 6.2. Here is another kind of weighted sum formula on multiple zeta values can be found in[20, Main Theorem]: For a pair of positive integers n and k with n ≥ k and k even, we have|α|=n (2 α2 + 2 α4 + • • • + 2 α k )ζ(α 1 , α 2 , . .…”