Combinatorial Implications of Decomposition Theorems of Multiple Zeta Values

摘要

The shuffle product of two multiple zeta values of weight m and n will produce (_m~(m+n)) multiple zeta values of weight m+n. The well-known Euler decomposition theorem can be obtained from the shuffle product of two Riemann zeta values. If we count the number of double Euler sums appearing in the identity, we obtain the well-known Pascal identity. Now in light of decomposition theorems of products of multiple zeta values of height one, we are able to produce several combinatorial identities which are generalizations of Pascal identity. Such as where d_1 + d_12 +...+ d_n =γ, with d_i ≥ 1, and S_n is the permutation group of n objects.