The algebra of Kleene stars of the plane and polylogarithms

We extend the definition and study the algebraic properties of the polylogarithm Li(T) , where T is rational series over the alphabet X = {x 0 , x 1 } belonging to suitable subalgebras of rational series.

[1]  A. Goncharov,et al.  Multiple polylogarithms and mixed Tate motives , 2001 .

[2]  Jianqiang Zhao,et al.  ANALYTIC CONTINUATION OF MULTIPLE ZETA FUNCTIONS , 1999 .

[3]  Li Guo,et al.  Differential Birkhoff decomposition and the renormalization of multiple zeta values , 2007, 0710.0432.

[4]  Li Guo,et al.  Renormalization of multiple zeta values , 2006 .

[5]  Vincel Hoang Ngoc Minh,et al.  On a conjecture by Pierre Cartier about a group of associators , 2009, 0910.1932.

[6]  Karol A. Penson,et al.  Mathematical Renormalization in Quantum Electrodynamics via Noncommutative Generating Series , 2015, 1702.08550.

[7]  Michel Petitot,et al.  Lyndon words, polylogarithms and the Riemann Zeta function , 2000, Discret. Math..

[8]  Hirofumi Tsumura,et al.  Desingularization of multiple zeta-functions of generalized Hurwitz-Lerch type , 2014, 1404.4758.

[9]  Gérard Jacob,et al.  Symbolic integration of meromorphic differential systems via Dirichlet functions , 2000, Discret. Math..

[10]  Hoang Ngoc Minh.  Summations of polylogarithms via evaluation transform , 1996 .

[11]  Sylvie Paycha,et al.  Nested sums of symbols and renormalized multiple zeta values , 2010 .

[12]  Christophe Tollu,et al.  (Pure) transcendence bases in φ-deformed shuffle bialgebras , 2015, ArXiv.

[13]  Jean Berstel,et al.  Rational series and their languages , 1988, EATCS monographs on theoretical computer science.

[14]  Michel Petitot,et al.  Combinatorial aspects of polylogarithms and Euler-Zagier sums. (Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier.) , 1999 .

[15]  C. Reutenauer.  Free Lie Algebras , 1993 .

[16]  Sylvie Paycha,et al.  Nested sums of symbols and renormalised multiple zeta functions , 2007 .

[17]  Christophe Tollu,et al.  Sweedler's duals and Schützenberger's calculus , 2007, ArXiv.

[18]  Gérard Duchamp,et al.  Harmonic sums and polylogarithms at negative multi-indices , 2015, ACCA.

[19]  Vincel Hoang Ngoc Minh,et al.  Noncommutative algebra, multiple harmonic sums and applications in discrete probability , 2009, J. Symb. Comput..