Generalizations of Euler decomposition and their applications

Abstract

The classical Euler decomposition theorem expressed a product of two Riemann zeta values in terms of double Euler sums. Such kind of decomposition theorem are useful to produce weighted sum formulae on double Euler sums. In this paper, we produce a more general decomposition theorem as a generalization of Euler decomposition theorem. Indeed, we obtained the shuffle product formula of the product ζ ( { 1 } j , m + r − ℓ + 2 ) ζ ( { 1 } k − j , q − m + ℓ + 2 ) with 0 ⩽ j ⩽ k , 0 ⩽ m ⩽ q and 0 ⩽ ℓ ⩽ r . Along with some well-known theorems from the theory of probability, we are able to build up a lot of double weighted sum formulae which express sums of multiple zeta values in terms of single zeta values. Such as ∑ c + d = k ∑ m = 0 q ∑ n = 0 r ∑ | α | = c + m + 1 , | β | = d + q − m + r + 2 , β d + 1 = q − m + n + 1 ζ ( α 0 , … , α c + β 0 , … , β d , β d + 1 + 1 ) ( β d + 1 − 1 n ) ( − 1 ) m + ∑ a + b = q ∑ | α | = k + q + r + 3 , α k + 1 = r + b + 2 ζ ( α 0 , … , α k , α k + 1 + 1 ) ( r + b r ) { ( − 1 ) a + ( − 1 ) q + r } = 1 2 ∑ j = 0 k ∑ m = 0 q ∑ ℓ = 0 r ( − 1 ) j + m + ℓ ζ ( { 1 } j , m + r − ℓ + 2 ) ζ ( { 1 } k − j , q − m + ℓ + 2 ) , where k + q + r is even.