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\titlehead{A combinatorial property of multiple polylogarithms at non-positive indices}
\authorhead{K. Kitamura}
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\title{A combinatorial property of multiple polylogarithms at non-positive indices}

\author{K. Kitamura \\
Department of Mathematics \\
The University of Osaka \\
Osaka, Japan, 560-0043 \\
\url{u347758b@ecs.osaka-u.ac.jp}}

\date{}

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\begin{abstract}
The (double) shuffle relations for multiple polylogarithms at positive indices are well-known and have very beautiful properties. On the other hand, there are some analogy such as \cite{Du17a},\cite{Du17b} and \cite{FMS} for MPL at non-positive (or general) indices. In this talk, we will show a new formula on products of MPL at non-positive indices in view of \cite{Na}, and we will give some applications.
\end{abstract}
\begin{thebibliography}{99}

\bibitem[1]{Du17a}
G. H. E. Duchamp, V. H. N. Minh, Q. H. Ngo, Harmonic sums and polylogarithms at non-positive multi indices, \textit{Journal of Symbolic Computation}, \textbf{83}(2017)166-186, 2016,\\
\url{https://doi.org/10.1016/j.jsc.2016.11.010}.
\bibitem[2]{Du17b}
G. H. E. Duchamp, V. H. N. Minh, Q. H. Ngo, K. A. Penson, P. Simonnet, Mathematical renormalization in quantum electrodynamics via noncommutative generating series, \textit{Applications of computer algebra}, 59–100, 2017, \\
\url{https://doi.org/10.48550/arXiv.1702.08550}.
\bibitem[3]{FMS}
Ebrahimi-Fard, Kurusch and Manchon, Dominique and Singer, Johannes, The Hopf Algebra of ($q$-)Multiple Polylogarithms with Non-positive Arguments, \textit{International Mathematics Research Notices}, \textbf{16}(2017) 4882-4922, 2016, \\
\url{https://doi.org/10.48550/arXiv.1503.02977}.
\bibitem[4]{Na}
H. Nakamura, Demi-shuffle duals of Magnus polynomials in a free associative algebra, \textit{Algebraic Combinatroics}, \textbf{6}(2023) no.4 929-939, 2021, \\
\url{https://doi.org/10.5802/alco.287}.

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