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In this paper, we define a finite sum analogue of multiple polylogarithms inspired by the work of Kaneko and Zaiger [KZ] and prove that they satisfy a certain analogue of the shuffle relation. Our result is obtained by using a certain partial fraction decomposition due to Komori-Matsumoto-Tsumura [KMT]. As a corollary, we give an algebraic interpretation of our shuffle product.

We define finite multiple zeta values (FMZVs) associated with some combinatorial objects, which we call 2-colored rooted trees, and prove that FMZVs associated with 2-colored rooted trees satisfying certain mild assumptions can be written explicitly as Z-linear combinations of the usual FMZVs. Our result can be regarded as a generalization of Kamano's recent work [K] on finite Mordell-Tornheim multiple zeta values. As an application, we will give a new proof of the shuffle relation of FMZVs, which was first proved by Kaneko and Zagier.

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