Publications Mathématiques de Besançon

Algèbre et Théorie des Nombres
Families of eulerian functions involved in regularization of divergent polyzetas
[Familles de fonctions eulériennes impliquées dans la régularisation de polyzêtas divergents]
V. C. Bui1 ; V. Hoang Ngoc Minh2 ; Q. H. Ngo3 ; V. Nguyen Dinh4
1 Hue University, 77, Nguyen Hue, Hue, Viet Nam,
2 University of Lille, 1 Place Déliot, 59024 Lille, France, LIPN - UMR 7030, CNRS, 93430 Villetaneuse, France
3 Hanoi University of Science and Technology, 1 Dai Co Viet, Hai Ba Trung, Ha Noi, Viet Nam,
4 LIPN-UMR 7030, 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse, France,
Publications mathématiques de Besançon. Algèbre et théorie des nombres (2023), pp. 5-28.

En généralisant les fonctions euleriennes, nous étudions leurs relations avec la fonction zêta en plusieurs variables. En particulier, à partir du théorème de factorisation de Weierstrass (et l’identité de Newton-Girard) pour la fonction Gamma complexe, nous nous intéressons aux rapports ζ(2k)/π 2k et leurs généralisations. Nous obtenons une situation analogue et nous tirerons quelques conséquences sur une structure de l’algèbre des valeurs polyzêtas, au moyen de la combinatoire des mots et des séries rationnelles en variables non commutatifs. Le même cadre de travail permet également d’étudier l’indépendance d’une famille de fonctions euleriennes.

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 ζ(2k)/π 2k and their multiindexed generalization, we obtain an analogue situation and draw some consequences about a structure of the algebra of polyzetas values, by means of some combinatorics of words and noncommutative rational series. The same frameworks also allow to study the independence of a family of eulerian functions.

Publié le :
DOI : 10.5802/pmb.47
Classification : 05E16, 11M32, 16T05, 20F10, 33F10, 44A20
Mots clés : Eulerian functions, zeta function, Gamma function
V. C. Bui 1 ; V. Hoang Ngoc Minh 2 ; Q. H. Ngo 3 ; V. Nguyen Dinh 4

1 Hue University, 77, Nguyen Hue, Hue, Viet Nam,
2 University of Lille, 1 Place Déliot, 59024 Lille, France, LIPN - UMR 7030, CNRS, 93430 Villetaneuse, France
3 Hanoi University of Science and Technology, 1 Dai Co Viet, Hai Ba Trung, Ha Noi, Viet Nam,
4 LIPN-UMR 7030, 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse, France,
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Families of eulerian functions involved in regularization of divergent polyzetas},
     journal = {Publications math\'ematiques de Besan\c{c}on. Alg\`ebre et th\'eorie des nombres},
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V. C. Bui; V. Hoang Ngoc Minh; Q. H. Ngo; V. Nguyen Dinh. Families of eulerian functions involved in regularization of divergent polyzetas. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2023), pp. 5-28. doi : 10.5802/pmb.47. https://pmb.centre-mersenne.org/articles/10.5802/pmb.47/

[1] Jean Berstel; Christophe Reutenauer Rational series and their languages, EATCS Monographs on Theoretical Computer Science, 12, Springer, 1988 | Zbl

[2] V. C. Bui; Gérard H. E. Duchamp; Q. H. Ngô; Vincel Hoang Ngoc Minh A local Theory of Domains and its (Noncommutative) Symbolic Counterpart, Math. Comput. Sci., Volume 17 (2023), 4 | DOI

[3] Christian Costermans; Vincel Hoang Ngoc Minh Noncommutative algebra, multiple harmonic sums and applications in discrete probability, J. Symb. Comput., Volume 44 (2009) no. 7, pp. 801-817 | DOI | MR | Zbl

[4] Matthieu Deneufchâtel; Gérard H. E. Duchamp; Vincel Hoang Ngoc Minh; Allan I. Solomon Independence of hyperlogarithms over function fields via algebraic combinatorics, Algebraic informatics, CAI 2011 (Linz, 2011) (Lecture Notes in Computer Science), Volume 6742, Springer, 2011, pp. 127-139 | MR | Zbl

[5] Jean Dieudonné Infinitesimal calculus, Houghton Mifflin, 1971

[6] Gérard H. E. Duchamp; Vincel Hoang Ngoc Minh; Ngo Quoc Hoan Kleene stars of the plane, polylogarithms and symmetries, Theor. Comput. Sci., Volume 800 (2019), pp. 52-72 | DOI | MR | Zbl

[7] Gérard H. E. Duchamp; Vincel Hoang Ngoc Minh; Vu Nguyen Dinh Towards a noncommutative Picard-Vessiot theory (2020) (https://arxiv.org/abs/2008.10872)

[8] Gérard H. E. Duchamp; Vincel Hoang Ngoc Minh; Karol A. Penson About Some Drinfel’d Associators, Computer algebra in scientific computing (Lille, 2018) (Lecture Notes in Computer Science), Volume 11077, Springer, 2018 | MR | Zbl

[9] Alexander B. Goncharov Multiple polylogarithms and mixed Tate motives (2001) (https://arxiv.org/abs/math/0103059v4)

[10] Vincel Hoang Ngoc Minh Summations of Polylogarithms via Evaluation Transform, Math. Comput. Simul., Volume 42 (1996) no. 4-6, pp. 707-728 | DOI | MR | Zbl

[11] Vincel Hoang Ngoc Minh Differential Galois groups and noncommutative generating series of polylogarithms, Automata, Combinatorics and Geometry (Systemics, Cybernetics and Informatics), 2003

[12] Vincel Hoang Ngoc Minh Finite polyzêtas, Poly-Bernoulli numbers, identities of polyzêtas and noncommutative rational power series, Proceedings of 4 th International Conference on Words (TUCS General Publication), Volume 27, Turku Centre for Computer Science, 2003, pp. 232-250 | MR | Zbl

[13] Vincel Hoang Ngoc Minh On the solutions of universal differential equation with three singularities, Confluentes Math., Volume 11 (2019) no. 2, pp. 25-64 | DOI | Numdam | MR | Zbl

[14] Alain Lascoux Fonctions symétriques, Sémin. Lothar. Comb., Volume 8 (1983), B08f, 16 pages | Zbl

[15] Adrien-Marie Legendre Exercices de calcul intégral sur divers ordres de transcendantes et sur les quadratures, 1, Courcier, 1811

[16] Marius van der Put; Michael F. Singer Galois Theory of Linear Differential Equations, Grundlehren der Mathematischen Wissenschaften, 328, Springer, 2003 | DOI

[17] Christophe Reutenauer Free Lie Algebras, London Mathematical Society Monographs. New Series, 7, Clarendon Press, 1993

[18] Jianqiang Zhao Analytic continuation of multiple zeta functions, Proc. Am. Math. Soc., Volume 128 (1999) no. 5, pp. 1275-1283 | DOI | MR | Zbl

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