A new proof of the duality of multiple zeta values and its generalizations

Abstract

We give a new proof of the duality of multiple zeta values, which makes no use of the iterated integrals. The same method is also applicable to Ohno's relation for (q-)multiple zeta values.2010 Mathematics Subject Classification. 11M32, 11B65.

Ohno-type identities for multiple harmonic sums

“…The case e = 0 gives Bradley's identity H ⋆ N (k; q) = z ⋆ N (k ∨ ; q). We will prove (4) by using a certain connected sum in §3, based on the same idea used in another paper of the authors [17]. This proof is new even if one specializes it to Hoffman's identity.…”

“…Proof. 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.…”

“…We call it a connected sum with connector C(m r , n 1 , q, x). In [17], another type of connected sums is used to give a new proof of Ohno's relation for the multiple zeta values and Bradley's q-analog of it.…”

Multivariable connected sums and multiple polylogarithms

“…The right-hand side of (1.1) is a special case of the connected sum introduced by the third author and Yamamoto [12]. We provide a brief review of this theory here.…”

“…We set ∅ → :=(1) and ∅ ↑ :=∅. Then, the transport relations for Z(k; l) are as follows (the q → 1, x → 0 case of [12,Theorem 2.2]):…”