Generalizations of Euler decomposition and their applications

Applications of shuffle product to restricted decomposition formulas for multiple zeta values

“…The appearance of the formula is different from the one in [9]. They should agree with each other after rearrangement of terms since both formulas are based the shuffle product.…”

“…This gives a so called decomposition formula when the product aXb on the right hand side is explicitly calculated even though a decomposition formula can also mean a formula derived from a process equivalent to the shuffle product, such as the one obtained in [9]. For example, when a = x m 0 x 1 and b = x n 0 x 1 this gives the Euler decomposition formula.…”

“…Euler's decomposition formula was generalized in [4,34] to the product of two q-zeta values. More recently, Euler's decomposition formula for multi-variable MZVs were obtained in [9,17]. In [17], Euler's decomposition formula is generalized to the product of any two MZVs by the algebraic method of double shuffle product.…”

“…In [17], Euler's decomposition formula is generalized to the product of any two MZVs by the algebraic method of double shuffle product. By an analytic method, Euler's decomposition formula is generalized in [9] to the product of two MZVs of the form ζ(m, {1} k ) with one string of 1's: {1} k := 1, · · · , 1 k terms . A formula for MZVs with strings of 1's is often called a restricted decomposition formula (see Section 3.1 for a general discussion on restricted decomposition formulas).…”

“…One is for MZVs with one strings of 1's similar to the one in [9]. The other one is for MZVs with one and two strings of 1's.…”