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Using some transformation formulas of the generalized hypergeometric series 3F2, we give another proof of D. Zagier’s evaluation formula of the multiple zeta values ζ(2, . . . , 2, 3, 2, . . . , 2). In a recent paper [5], Zagier found and proved an evaluation formula ζ(2, . . . , 2 } {{ } a , 3, 2, . . . , 2 } {{ } b ) = 2 a+b+1 ∑ r=1 (−1)ca,bζ(2r + 1)ζ(2, . . . , 2 } {{ } a+b+1−r ), (1) where a, b are nonnegative integers and ca,b = ( 2r 2a+ 2 ) − ( 1− 1 22r )( 2r 2b+ 1 ) . Here for positive integers k1, . . . , kn with kn 2, the multiple zeta value ζ(k1, . . . , kn) is defined by the following infinite series: ζ(k1, . . . , kn) = ∑ 0<m1<···<mn 1 m1 1 · · ·mn n . The evaluation formula (1) plays an important role in the work of Brown [2], who showed that all multiple zeta values can be represented as Q-linear combinations of multiple zeta values of the same weight with all arguments are 2’s and 3’s and all periods of mixed Tate motives over Z are Q[(2πi)±1]-linear combinations of multiple zeta values. As in [5], let H(a, b) (resp. Ĥ(a, b)) denote the left-hand side (resp. the right-hand side) of (1). One considers the following two generating functions:

Another proof of Zagier’s evaluation formula of the multiple zeta values $\zeta (2, \dots , 2, 3, 2, \dots , 2)$