Weighted sum formulas from shuffle products of multiples of Riemann zeta values

Abstract

The classical Euler decomposition theorem expressed a product of two Riemann zeta values in terms of double Euler sums. It can also be obtained from the shuffle product of two Riemann zeta values when both are expressed as iterated integrals due to Kontsevich. In this paper, we investigate the shuffle product of n multiples of Riemann zeta values(cj+1)ζ(cj+2),j=1,2,…,n,cj≥0 through their particular integral representations. Among other things, we obtain the following weighted sum formula∑|α|=k∑βζ(α1+β1+1,α2+β2+1,…,αn+βn+1)∏j=1n(αj+βj)!αj!jαj=1n!∑|c|=k∏j=1n(cj+1)ζ(cj+2) with∑βx1β1x2β2⋯xnβn=(x1+x2+⋯+xn)(x2+⋯+xn)⋯xn which is a sum of n! monomials of degree n. Here ζ(a1,a2,…,an) appearing in the summation is a multiple zeta value or n-fold Euler sum. Also we obtain the weighted sum formula of triple Euler sums∑|α|=k+5ζ(α1,α2,α3+1)⋅{2α2−1(3α3−2α3)−(2α3−1)}−∑|α|=k+4ζ(1,α2,α3+1)⋅(2α3−1)+ζ(1,1,k+4)=16∑|c|=kζ(c1+2)ζ(c2+2)ζ(c3+2).