Kleene stars of the plane, polylogarithms and symmetries

A local Theory of Domains and its (Noncommutative) Symbolic Counterpart

“…One way to obtain relations among polyzetas is to consider their generating series and the relations among the coefficients of these generating series. This leads to the re-indexation of the coefficients of the generating series of polylogarithms itself, recently mentioned in [5,13] and, in a way, this work is a continuation of [6]. But, in order to understand the bridge between this extension of this "polylogarithmic calculus" and the world of harmonic sums, a local theory of domains has to be done, preserving quasi-shuffle identities, Taylor expansions and Hadamard products.…”

“…This work is partly the continuation of [6,5,13] where it has been established that the polylogarithms, indexed by the r-tuples (s 1 , . .…”

“…To this end a theory of global domains was presented in [6,5,13]. Here we focus on what happens in the neighbourhood of zero, therefore, the aim of this work is manyfold.…”

“…(iii) Taking the definition of polyetas as in (1) at z = 1 or in (6) at +∞, one sees that, for any s 1 > 1, Abel's theorem, one has ζ (s 1 , . .…”

On the solutions of the universal differential equation with three regular singularities (On solutions ofKZ3)

“…[22,38,41]). -(The operators { 0 , 1 , 0 , 1 } satisfy 1 )( 1 0 ) = ( 1 0 )( 0 1 ) = Id.The subspace {Li } ∈ * is closed under the action of { 0 , 1 } and { 0 , 1 }.…”