A restricted sum formula among multiple zeta values

Abstract

For positive integers α 1 , α 2 , . . . , α r with α r 2, the multiple zeta value or r-fold Euler sum is defined asThere is a celebrated sum formula among multiple zeta values as |α|=m ζ(α 1 , α 2 , . . . , α r + 1) = ζ(m + 1), where α 1 , α 2 , . . . , α r range over all positive integers with |α| = α 1 + α 2 + · · · + α r = m in the summation. In this paper, we shall prove that for all positive integers m and q with m q, and a nonnegative integer p,When p = 0 and q = r, this is precisely the sum formula. Such a formula can be used to determine explicitly, some multiple zeta values of lower weights more precisely than the sum formula.

A short proof for the sum formula and its generalization

“…and give another proof of the following generalization of the sum formula which originally due to the first author [4]. …”

“…. , αr) = 1≤k 1 <k 2 <···<krThere is a celebrated sum formula [6, 10] among multiple zeta values aswhere α1, α2, ..., αr range over all positive integers with |α α α| = α1 + α2 + · · · + αr = m in the summation.In this paper, we shall prove the so called restricted sum formula [4]. Namely, for all positive integers m and q with m ≥ q and a nonnegative integer p, that α i ≥1, ∀i |α α α|=m ζ({1} p , α1, α2, .…”

“…In this paper, we shall prove the so called restricted sum formula [4]. Namely, for all positive integers m and q with m ≥ q and a nonnegative integer p, that α i ≥1, ∀i |α α α|=m ζ({1} p , α1, α2, .…”

Back Matter

“…Multiple zeta values [23,25,44,52] are natural generalizations of Euler sums. For positive integers α 1 , α 2 , .…”

Some alternating double sum formulae of multiple zeta values

“…Multiple zeta values are multidimensional version of the Euler sums [1,6,9,13,14,15]. For a string of positive integers α = (α 1 , α 2 , .…”