On the solutions of the universal differential equation with three regular singularities (On solutions of KZ 3 )
Confluentes Mathematici, Volume 11 (2019) no. 2, pp. 25-64.

This review concerns the resolution of a special case of Knizhnik-Zamolodchikov equations (KZ 3 ) and our recent results on combinatorial aspects of zeta functions on several variables.

In particular, we describe the action of the differential Galois group of KZ 3 on the asymptotic expansions of its solutions leading to a group of associators which contains the unique Drinfel’d associator (or Drinfel’d series). Non trivial expressions of an associator with rational coefficients are also explicitly provided, based on the algebraic structure and the singularity analysis of the multi-indexed polylogarithms and harmonic sums.

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DOI: 10.5802/cml.59
Classification: 05E16, 11M32, 16T05, 20F10, 33F10, 44A20
Keywords: Algebraic Basis, Combinatorial Hopf Algebra, Harmonic Sum, Polylogarithm, Polyzeta
Vincel Hoang Ngoc Minh 1

1 LIPN - UMR 7030, CNRS, 93430 Villetaneuse, France. University of Lille, 1 Place Déliot, 59024 Lille, France.
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Vincel Hoang Ngoc Minh. On the solutions of the universal differential equation with three regular singularities (On solutions of $KZ_3$). Confluentes Mathematici, Volume 11 (2019) no. 2, pp. 25-64. doi : 10.5802/cml.59. https://cml.centre-mersenne.org/articles/10.5802/cml.59/

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