ON THE SOLUTIONS OF THE UNIVERSAL DIFFERENTIAL EQUATION WITH THREE REGULAR SINGULARITIES (ON SOLUTIONS OF 3 )

Vincel Hoang Ngoc Minh (1, 2)

1 Université de Lille, Sciences et Technologies
2 LIPN

This review concerns the resolution of a special case of Knizhnik-Zamolodchikov equations (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 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. Contents 1. Knizhnik-Zamolodchikov equations and Drinfel'd series 2. Combinatorial framework 2.1. Shuffle and quasi-shuffle algebras 2.2. Diagonal series on bialgebras 2.3. Exchangeable and noncommutative rational series 3. Indexation by words and generating series 3.1. Indexation by words 3.2. Indexation by noncommutative rational series 3.3. Noncommutative generating series 4. Global asymptotic behaviors at singularities 4.1. The case of positive multi-indices 4.2. Structure of polyzetas 4.3. The case of negative multi-indices 5. A group of associators 5.1. The action of the Galois differential group 5.2. Associator Φ 5.3. Associators with rational coefficients 6. Conclusion Appendix A Appendix B Appendix C Appendix D References Math. classification: ??

Mots clés: Algebraic Basis; Combinatorial Hopf Algebra; Harmonic Sum; Polylogarithm; Polyzeta

Domaines: Mathématiques [math]; Informatique [cs]

Vincel Hoang Ngoc Minh. ON THE SOLUTIONS OF THE UNIVERSAL DIFFERENTIAL EQUATION WITH THREE REGULAR SINGULARITIES (ON SOLUTIONS OF 3 ). Confluentes Mathematici, 2020. ⟨hal-03558929⟩