On the solutions of the universal differential equation with three regular singularities (On solutions of $K{Z}_3$)

Hoang Ngoc Minh, Vincel

Confluentes Mathematici, Tome 11 (2019) no. 2, pp. 25-64.

This review concerns the resolution of a special case of Knizhnik-Zamolodchikov equations ($K{Z}_3$) 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 $K{Z}_3$ 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.

Classification : 05E16, 11M32, 16T05, 20F10, 33F10, 44A20
Mots clés : Algebraic Basis, Combinatorial Hopf Algebra, Harmonic Sum, Polylogarithm, Polyzeta

[1] J. Berstel, C. Reutenauer. Rational series and their languages, Springer Verlag, 1988. | Zbl

[2] M. Bigotte. Etude symbolique et algorithmique des fonctions polylogarithmes et des nombres d’Euler-Zagier colorés, Ph. D., Lille, 2000.

[3] V.C. Bui, G.H.E. Duchamp, V. Hoang Ngoc Minh. Schützenberger’s factorization on the (completed) Hopf algebra of $q-$stuffle product, Journal of Algebra, Number Theory and Applications, pp. 191-215, 30, No. 2, 2013. | Zbl

[4] V.C. Bui, G.H.E. Duchamp, V. Hoang Ngoc Minh, L. Kane, C. Tollu. Dual bases for non commutative symmetric and quasi-symmetric functions via monoidal factorization, dans Journal of Symbolic Computation (2015). | DOI | MR | Zbl

[5] V.C. Bui, G.H.E. Duchamp, V. Hoang Ngoc Minh. Structure of Polyzetas and Explicit Representation on Transcendence Bases of Shuffle and Stuffle Algebras, Journal of Symbolic Computation, 83, pp. 93-111 (2017). | DOI | MR | Zbl

[6] V.C. Bui, G.H.E. Duchamp, N. Hoang, V. Hoang Ngoc Minh, C. Tollu. Combinatorics on the $\phi$-deformed stuffle product, arXiv:1302.5391.

[7] V.C. Bui, G.H.E. Duchamp, V. Hoang Ngoc Minh, Q.H. Ngo, C. Tollu. (Pure) transcendence bases in $\varphi$-deformed shuffle bialgebras, Journal Electronique du Séminaire Lotharingien de Combinatoire, 74 (2018). | MR | Zbl

[8] N. Bourbaki. Topologie générale, Springer (1974) | Zbl

[9] P. Cartier. Jacobiennes généralisées, monodromie unipotente et intégrales itérées, Séminaire Bourbaki, 687 (1987), 31–52.

[10] P. Cartier. Fonctions polylogarithmes, nombres polyzetas et groupes pro-unipotents. Séminaire Bourbaki, ${53}^{ème}$, $n^{\circ }885$, 2000-2001.

[11] K.-T. Chen. Iterated integrals and exponential homomorphisms, Proc. Lond. Mathem. Soc. (3) 4 (1954) 502–512. | DOI | MR | Zbl

[12] C. Costermans. Calcul non nommutatif : analyse des constantes d’arbre de fouille, thèse, Lille, 2008.

[13] C. Costermans, J.Y. Enjalbert and V. Hoang Ngoc Minh. Algorithmic and combinatoric aspects of multiple harmonic sums, Discrete Mathematics & Theoretical Computer Science Proceedings, 2005. | Zbl

[14] C. Costermans, V. Hoang Ngoc Minh. Some Results à l’Abel Obtained by Use of Techniques à la Hopf, “Workshop on Global Integrability of Field Theories and Applications”, Daresbury (UK), 1-3, November 2006. | Zbl

[15] C. Chevalley. Fundamental Concepts of Algebra, Academic Press, New York, 1956. | Zbl

[16] C. Costermans, V. Hoang Ngoc Minh. Noncommutative algebra, multiple harmonic sums and applications in discrete probability, Journal of Symbolic Computation, (2009), 801-817. | DOI | MR | Zbl

[17] M. Deneufchâtel. Intégrales itérées et physique combinatoire, Ph. D., Paris 13, 2012.

[18] P. Deligne. Equations Différentielles à Points Singuliers Réguliers, Lecture Notes in Math, 163, Springer-Verlag (1970). | DOI | Zbl

[19] M. Deneufchâtel, G.H.E. Duchamp, V. Hoang Ngoc Minh, A.I. Solomon. Independence of hyperlogarithms over function fields via algebraic combinatorics, dans Lec. N. in Comp. Sc. (2011), Volume 6742/2011, 127-139. | DOI | Zbl

[20] J. Dieudonné. Calcul infinitésimal, Hermann (1968). | DOI

[21] G.H.E. Duchamp, V. Hoang Ngoc Minh, Q.H. Ngo. Harmonic sums and polylogarithms at negative multi-indices, Journal of Symbolic Computation, 83, pp. 166-186 (2017). | DOI | MR | Zbl

[22] G.H.E. Duchamp, V. Hoang Ngoc Minh, Q.H. Ngo. Kleene stars of the plane, polylogarithms and symmetries, soumitted to TCS. | DOI | MR | Zbl

[23] V. Drinfel’d. Quantum groups, Proc. Int. Cong. Math., Berkeley, 1986.

[24] V. Drinfel’d. Quasi-Hopf Algebras, Len. Math. J., 1, 1419-1457, 1990.

[25] V. Drinfel’d. On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal($\overline{\mathbb{Q}}/\mathbb{Q}$), Leningrad Math. J., 4, 829-860, 1991. | Zbl

[26] F.J. Dyson, The radiation theories of Tomonaga, Schwinger and Feynman, Physical Rev, vol 75, (1949), pp. 486-502. | DOI | MR | Zbl

[27] S. Eilenberg and S. Mac Lane. On the groups H($\Pi$, n) I, Ann. of Math. 58 (1) (1953) 55–106. | DOI

[28] E. Wardi. Mémoire de DEA, Lille, 1999.

[29] J.Y. Enjalbert, V. Hoang Ngoc Minh. Analytic and combinatorial aspects of Hurwitz polyzêtas, J. Th. des Nomb. de Bord., Vol. 19, (2007), 599-644. | DOI | MR | Zbl

[30] J.Y. Enjalbert, V. Hoang Ngoc Minh. Propriétés combinatoires et prolongement analytique effectif de polyzêtas de Hurwitz et de leurs homologues, J. Th. des Nomb. de Bord., 23 no. 2 (2011), 353-386. | DOI | Zbl

[31] J.Y. Enjalbert, G.H.E. Duchamp, V. Hoang Ngoc Minh, C. Tollu. The contrivances of shuffle products and their siblings, Discrete Mathematics 340(9): 2286-2300 (2017). | DOI | MR | Zbl

[32] P. Flajolet, R. Sedgewick. Analytic combinatorics, Cambridge University Press, 2009. | DOI | Zbl

[33] M. Fliess. Fonctionnelles causales non linéaires et indéterminées non commutatives, Bull. Soc. Math. France, N${}^{\circ }$109, 1981, pp. 3-40. | DOI | Zbl

[34] J. Gonzalez-Lorca. Série de Drinfel’d, monodromie et algèbres de Hecke, Ph. D., Ecole Normale Supérieure, Paris, 1998.

[35] V Hoang Ngoc Minh. Summations of Polylogarithms via Evaluation Transform, dans Math. & Computers in Simulations, 1336, pp 707-728, 1996. | DOI | MR | Zbl

[36] V. Hoang Ngoc Minh. Fonctions de Dirichlet d’ordre $n$ et de paramètre $t$, Disc. Math. 180, pp 221-242, 1998. | DOI | MR

[37] V. Hoang Ngoc Minh & Jacob G.. Symbolic Integration of meromorphic differential equation via Dirichlet functions, Disc. Math. 210, pp. 87-116, 2000. | DOI | MR | Zbl

[38] V. Hoang Ngoc Minh, G. Jacob, N.E. Oussous, M. Petitot. Aspects combinatoires des polylogarithmes et des sommes d’Euler-Zagier, Journal Electronique du Séminaire Lotharingien de Combinatoire, B43e, (2000).

[39] V. Hoang Ngoc Minh, G. Jacob, N.E. Oussous, M. Petitot. De l’algèbre des $\zeta$ de Riemann multivariées à l’algèbre des $\zeta$ de Hurwitz multivariées, Journal Electronique du Séminaire Lotharingien de Combinatoire, 44, (2001).

[40] V. Hoang Ngoc Minh, M. Petitot. Lyndon words, polylogarithmic functions and the Riemann $\zeta$ function, Discrete Math., 217, 2000, pp. 273-292. | DOI | MR | Zbl

[41] V. Hoang Ngoc Minh, M. Petitot, J. Van der Hoeven. Polylogarithms and Shuffle Algebra, Proceedings of FPSAC’98, 1998.

[42] V. Hoang Ngoc Minh. Finite polyzêtas, Poly-Bernoulli numbers, identities of polyzêtas and noncommutative rational power series, Proceedings of $4^{th}$ International Conference on Words, pp. 232-250, 2003. | MR | Zbl

[43] V. Hoang Ngoc Minh. Differential Galois groups and noncommutative generating series of polylogarithms, Automata, Combinatorics & Geometry, World Multi-conf. on Systemics, Cybernetics & Informatics, Florida, 2003.

[44] V. Hoang Ngoc Minh. Shuffle algebra and differential Galois group of colored polylogarithms, Nuclear Physics B, 135 (2004), pp. 220-224. | DOI | MR

[45] V. Hoang Ngoc Minh. Algebraic Combinatoric Aspects of Asymptotic Analysis of Nonlinear Dynamical System with Singular Inputs, Acta Academiae Aboensis, Ser. B, Vol. 67, no. 2, (2007), pp. 117-126.

[46] V. Hoang Ngoc Minh. On a conjecture by Pierre Cartier about a group of associators, Acta Math. Vietnamica (2013), 38, Issue 3, 339-398. | DOI | MR | Zbl

[47] V. Hoang Ngoc Minh. Structure of polyzetas and Lyndon words, Vietnamese Math. J. (2013), 41, Issue 4, 409-450. | DOI | MR | Zbl

[48] M. Hoffman. Quasi-shuffle products, J. Alg. Combin. 11 (1) (2000) 49–68. | MR | Zbl

[49] D. Knutson. $\lambda$-rings and the representation theory of the symmetric group, Lecture Notes in Mathematics, vol. 308, Springer-Verlag, 1973. | DOI | MR | Zbl

[50] J.A. Lappo-Danilevsky. Théorie des systèmes des équations différentielles linéaires, Chelsea, New York, 1953. | Zbl

[51] A. Lascoux. Fonctions symétriques, journal électronique du Séminaire Lotharingien de Combinatoire, B08e, (1983).

[52] T.Q.T. Lê, J. Murakami. Kontsevich’s integral for Kauffman polynomial, Nagoya Math., pp 39-65, 1996. | DOI | MR | Zbl

[53] M. Lothaire. Combinatorics on Words, Encyclopedia of Mathematics and its Applications, Addison-Wesley, 1983.

[54] W. Magnus. On the exponential solution of differential equations for a linear operator., AC on Pure and App. Math., VII:649–673, 1954. | DOI | MR | Zbl

[55] G. Racinet. Séries génératrices non-commutatives de polyzêtas et associateurs de Drinfel’d, Ph. D., Amiens, 2000.

[56] R. Ree. Lie elements and an algebra associated with shuffles Ann. Math 68 210–220, 1958. | DOI | MR | Zbl

[57] C. Reutenauer. Free Lie Algebras, London Math. Soc. Monographs (1993). | Zbl

[58] G. Viennot. Algèbres de Lie libres et monoïdes libres, Lecture Notes in Mathematics, Springer-Verlag, 691, 1978. | Zbl

[59] D. Zagier. Values of zeta functions and their applications, in “First European Congress of Mathematics”, vol. 2, Birkhäuser, pp. 497-512, 1994. | DOI | Zbl