Duality in Tails of Multiple-Zeta Values

Abstract

Duality relations are deduced for tails of multiple-zeta values using elementary methods. These formulas extend the classical duality formulas for multiple-zeta values.

Harmonic-number summation identities, symmetric functions, and multiple zeta values

“…. , a k ) of n, the monomial symmetric function M (a 1 ,...,a k ) is the "smallest" quasi-symmetric function containing x a 1 1 · · · x a k k ; for example, the formal power series (15) is M (2,1) . Any quasi-symmetric function of degree n is a linear combination of monomial quasi-symmetric functions of the same degree.…”

“…Let H (r) n denote the generalized harmonic number n j=1 1 n r ; if r = 1 we omit the superscript. This paper is concerned with series of the form ∞ n=1 F (H n , H (2) n , . .…”

“…, i k with i 1 > 1. For integer s ≥ 2, any sum of the form ∞ n=1 H (r) n n s , which includes Euler's examples (2), is readily expressible in terms of multiple zeta values as ζ(r + s) + ζ(s, r). In recent decades intensive study has led to the development of an extensive theory of multiple zeta values, which we summarize in §3 below.…”

“…(The Q k express the complete symmetric functions in terms of power sums.) Not only is the identity ∞ n=1 P k (H n , H (2) n , . .…”

“…for integers k > 0 and q ≥ 2, which generalizes the case l = 0 of Theorems 2 and 5. We note that the case k = 1 of identity (14) coincides with Theorem 1 of [2]. Our main technical tool is the introduction of a class of functions η s 1 ,...,s k , which we call H-functions, from the quasi-symmetric functions (a superalgebra of the symmetric functions) to the reals such that…”