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In this paper we consider iterated integral representations of multiple polylogarithm functions and prove some explicit relations of multiple polylogarithm functions. Then we apply the relations obtained to find numerous formulas of alternating multiple zeta values in terms of unit-exponent alternating multiple zeta values. In particular, we prove several conjectures given by Borwein-Bradley-Broadhurst [3] , and give some general results.

We generalize the definition of overconvergent $p$-adic multiple polylogarithms and of $p$-adic cyclotomic multiple zeta values and we prove a bound on their norm. A byproduct of the proof is a characterization of these objects in terms of certain regularized $p$-adic iterated integrals. The generalization of the definition consists in replacing the underlying Frobenius structure by its iterations. The bound on the norms of overconvergent $p$-adic multiple polylogarithms that we obtain is a prerequisite for our subsequent papers on $p$-adic cyclotomic multiple zeta values., 25 pages. This is Part I-1 of "$p$-adic cyclotomic multiple zeta values and $p$-adic pro-unipotent harmonic actions"

We study the algebra of certain q-series, called bi-brackets, whose coefficients are given by weighted sums over partitions. These series incorporate the theory of modular forms for the full modular group as well as the theory of multiple zeta values (MZV) due to their appearance in the Fourier expansion of regularized multiple Eisenstein series. Using the conjugation of partitions we obtain linear relations between bi-brackets, called the partition relations, which yield naturally two different ways of expressing the product of two bi-brackets similar to the stuffle and shuffle product of multiple zeta values. In a recent work, the author and K. Tasaka defined (shuffle) regularized multiple Eisenstein series G ⧢ , by using an explicit connection to the coproduct on formal iterated integrals introduced by Goncharov. These satisfy the shuffle product formula. Applying the same concept for the coproduct on quasi-shuffle algebras enables us to define (stuffle) regularized multiple Eisenstein series G ⁎ satisfying the stuffle product formula. We show that both G ⧢ and G ⁎ are given by linear combinations of products of MZV and bi-brackets.

We investigate linear relations among a class of iterated integrals on the Riemann sphere minus four points $0,1,z$ and $\infty$. Generalization of the duality formula and the sum formula for multiple zeta values to the iterated integrals are given., Comment: 11 pages

We prove a formula among finite multiple zeta values with four parameters. This formula is proved by using the iterated integral expression of the multiple polylogarithms. By specializing these parameters, we give some weighted sum formulas for finite multiple zeta values.