Related papers: New functional equations of finite multiple polylo…
We survey various results and conjectures concerning multiple polylogarithms and the multiple zeta function. Among the results, we announce our resolution of several conjectures on multiple zeta values. We also provide a new integral…
In this paper, we define a finite sum analogue of multiple polylogarithms inspired by the work of Kaneko and Zaiger and prove that they satisfy a certain analogue of the shuffle relation. Our result is obtained by using a certain partial…
The finite n-th polylogarithm li_n(z) in Z/p[z] is defined as the sum on k from 1 to p-1 of z^k/k^n. We state and prove the following theorem. Let Li_k:C_p to C_p be the p-adic polylogarithms defined by Coleman. Then a certain linear…
By using the method of iterated integral representations of series, we establish some explicit relationships between multiple zeta values and Integrals of logarithmic functions. As applications of these relations, we show that multiple zeta…
Maximon has recently given an excellent summary of the properties of the Euler dilogarithm function and the frequently used generalizations of the dilogarithm, the most important among them being the polylogarithm function $Li_(z)$. The…
In this paper we consider iterated integrals 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…
We define subvarieties of $\mathcal{M}_{0,n}$ equipped with algebraic functions that are solutions to the generic double shuffle equations satisfied by multiple polylogarithms on $\mathcal{M}_{0,n}$.
We introduce the multivariable connected sum which is a generalization of Seki-Yamamoto's connected sum and prove the fundamental identity for these sums by series manipulation. This identity yields explicit procedures for evaluating…
We consider the problem of finding the set of classical polylogarithmic functions $\text{Li}_n$ with branching locus determined by the solution of $p_1\cdot p_2\cdot \ldots \cdot p_n=0$, where $p_1,\ldots, p_n$ are irreducible polynomials…
We study a polynomial interpolation of finite multiple zeta and zeta-star values with variable $t$, which is an analogue of interpolated multiple zeta values introduced by Yamamoto. We introduce several relations among them and, in…
Recently, several people study finite multiple zeta values (FMZVs) and finite polylogarithms (FPs). In this paper, we introduce finite multiple polylogarithms (FMPs), which are natural generalizations of FMZVs and FPs, and we establish…
Polylogarithmic functions (polylogs) in $n$ variables can be viewed as elements of $(U\mathfrak{p}_{m})^*$, the dual of the universal enveloping algebra of the Lie algebra $\mathfrak{p}_{m}$ of infinitesimal spherical pure braids with…
An efficient procedure for the computation of $Li_{s}(z)$ where $s<0$ is here presented. We started with Polylogarithm $Li_{s}(z)$ where $s<0$. The summation of $n^{s}z^{n}$ is evaluated using a new method. An assumption is made that the…
We introduce two new bases for polynomials that lift monomial and fundamental quasisymmetric functions to the full polynomial ring. By defining a new condition on pipe dreams, called quasi-Yamanouchi, we give a positive combinatorial rule…
Functional equations, in the form of fusion hierarchies, are studied for the transfer matrices of the fused restricted $A_{n-1}^{(1)}$ lattice models of Jimbo, Miwa and Okado. Specifically, these equations are solved analytically for the…
In this paper we first establish several integral identities. These integrals are of the form \[\int_0^1 x^{an+b} f(x)\,dx\quad (a\in\{1,2\},\ b\in\{-1,-2\})\] where $f(x)$ is a single-variable multiple polylogarithm function or…
Kaneko and Tsumura introduced the Arakawa-Kaneko type zeta function $\eta(-k_1,\ldots,-k_r;s_1,\ldots,s_r)$ for non-negative integers $k_1,\ldots,k_r$ and complex variables $s_1,\ldots,s_r$. Recently, Yamamoto showed that, by using the…
Xu, Yan and Zhao showed that in even weight, the multiple $T$ value $T(2, 1, \ldots, 1, \overline{1})$ is a polynomial in $\log(2)$, $\pi$, Riemann zeta values, and Dirichlet beta values. Based on low-weight examples, they conjectured that…
We treat desingularized multiple zeta-functions introduced by Furusho, Komori, Matsumoto and Tsumura. In this paper, we prove functional relations, which are shuffle type product formulae, between desingularized multiple zeta-functions and…
We introduce and study new versions of polylogarithms and a zeta function on a completion of $\mathbb F_q (x)$ at a finite place. The construction is based on the use of the Carlitz differential equations for $\mathbb F_q$-linear functions.