EXTENDED DOUBLE SHUFFLE RELATIONS AND THE GENERATING FUNCTION OF TRIPLE ZETA VALUES OF ANY FIXED WEIGHT

Abstract

Extended double shuffle relations for multiple zeta values are obtained by using the fact that any product of regularized multiple zeta values has two different representations. In this paper, we give two formulas for the generating function of the triple zeta values of any fixed weight by the use of the extended double shuffle relations obtained as two-fold products of double and single zeta values and also as three-fold products of single zeta values. As applications of the formulas, we also obtain parameterized, weighted, and restricted sum formulas for triple zeta values.