1,621 research outputs found
On the solutions of universal differential equation by noncommutative Picard-Vessiot theory
Basing on Picard-Vessiot theory of noncommutative differential equations and
algebraic combinatorics on noncommutative formal series with holomorphic
coefficients, various recursive constructions of sequences of grouplike series
converging to solutions of universal differential equation are proposed. Basing
on monoidal factorizations, these constructions intensively use diagonal series
and various pairs of bases in duality, in concatenation-shuffle bialgebra and
in a Loday's generalized bialgebra. As applications, the unique solution,
satisfying asymptotic conditions, of Knizhnik-Zamolodchikov equations is
provided by d\'evissage
On The Global Renormalization and Regularization of Several Complex Variable Zeta Functions by Computer
This review concerns the resolution of a special case of
Knizhnik-Zamolodchikov equations () using our recent results on
combinatorial aspects of zeta functions on several variables and software on
noncommutative symbolic computations. In particular, we describe the actual
solution of leading to the unique noncommutative series, ,
so-called Drinfel'd associator (or Drinfel'd series). Non-trivial expressions
for series with rational coefficients, satisfying the same properties with
, are also explicitly provided due to the algebraic structure and
the singularity analysis of the polylogarithms and harmonic sums
Families of eulerian functions involved in regularization of divergent polyzetas
Extending the Eulerian functions, we study their relationship with zeta
function of several variables. In particular, starting with Weierstrass
factorization theorem (and Newton-Girard identity) for the complex Gamma
function, we are interested in the ratios of and their
multiindexed generalization, we will obtain an analogue situation and draw some
consequences about a structure of the algebra of polyzetas values, by means of
some combinatorics of noncommutative rational series. The same combinatorial
frameworks also allow to study the independence of a family of eulerian
functions.Comment: preprin
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