Abstract
We describe some particular finite sums of multiple zeta values which arise from J. Ecalle’s “arborification”, a process which can be described as a surjective Hopf algebra morphism from the Hopf algebra of decorated rooted forests onto a Hopf algebra of shuffles or quasi-shuffles. This formalism holds for both the iterated sum picture and the iterated integral picture. It involves a decoration of the forests by the positive integers in the first case, by only two colours in the second case.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Brown, F.: On the decomposition of motivic multiple zeta values. Galois-Teichmüller theory and arithmetic geometry. Adv. Stud. Pure Math. 63, 31–58 (2012)
Brown, F.: Mixed Tate motives over \({\mathbb{Z}}\). Ann. Math. 175(1), 949–976 (2012)
Brown, F.: Depth-graded motivic multiple zeta values, preprint, arXiv:1301.3053 (2013)
Butcher, J.C.: An algebraic theory of integration methods. Math. Comp. 26, 79–106 (1972)
Cartier, P.: Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents, Séminaire Bourbaki No 885. Astérisque 282, 137–173 (2002)
Connes, A., Kreimer, D.: Hopf Algebras, Renormalization and Noncommutative Geometry. Comm. Math. Phys. 199, 203–242 (1998)
Deligne, P., Goncharov, A.: Groupes fondamentaux motiviques de Tate mixtes. Ann. Sci. Ec. Norm. Sup. (4) 38(1), 1–56 (2005)
Dür, A.: Möbius functions, incidence algebras and power series representations, Lect. Notes math. 1202, Springer (1986)
Ecalle, J.: Les fonctions résurgentes Vol. 2, Publications Mathématiques d’Orsay (1981). Available at http://portail.mathdoc.fr/PMO/feuilleter.php?id=PMO_1981
Ecalle, J.: Singularités non abordables par la géométrie. Ann. Inst. Fourier 42(1–2), 73–164 (1992)
Ecalle, J., Vallet, B.: The arborification-coarborification transform: analytic, combinatorial, and algebraic aspects. Ann. Fac. Sci. Toulouse XIII(4), 575–657 (2004)
Eie, M., Liaw, W., Ong, Y.L.: A restricted sum formula for multiple zeta values. J. Number Theory 129, 908–921 (2009)
Fauvet, F., Menous, F.: Ecalle’s arborification-coarborification transforms and the Connes-Kreimer Hopf algebra. Ann. Sci. Ec. Norm. 50(1), 39–83 (2017)
Foissy, L.: Les algèbres de Hopf des arbres enracinés décorés I,II. Bull. Sci. Math. 126, 193–239, 249–288 (2002)
Hoffman, M.E.: Multiple harmonic series. Pacific J. Math. 152, 275–290 (1992)
Hoffman, M.E.: Quasi-shuffle products. J. Algebraic Combin. 11, 49–68 (2000)
Ihara, K., Kaneko, M., Zagier, D.: Derivation and double shuffle relations for multiple zeta values. Comp. Math. 142(2), 307–338 (2004)
Kaneko, M., Yamamoto, S.: A new integral-series identity of multiple zeta values and regularizations. Selecta Math. 24(3), 2499–2521 (2018)
Kreimer, D.: On the Hopf algebra structure of perturbative quantum field theories. Adv. Theor. Math. Phys. 2, 303–334 (1998)
Murua, A.: The Hopf algebra of rooted trees, free Lie algebras, and Lie series. Found. Computational Math. 6, 387–426 (2006)
Racinet, G.: Doubles mélanges des polylogarithmes multiples aux racines de l’unité. Publ. Math. IHES 95, 185–231 (2002)
Schmitt, W.: Incidence Hopf algebras. J. Pure Appl. Algebra 96, 299–330 (1994)
Tanaka, T.: Restricted sum formula and derivation relation for multiple zeta values. arXiv:1303.0398 (2014)
Terasoma, T.: Mixed Tate motives and multiple zeta values. Invent. Math. 149(2), 339–369 (2002)
Yamamoto, S.: Multiple zeta-star values and multiple integrals, preprint. arXiv:1405.6499 (2014)
Zagier, D.: Values of zeta functions and their applications. Proc. First Eur. Congress Math. 2, 497–512, Birkhäuser, Boston (1994)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Manchon, D. (2020). Arborified Multiple Zeta Values. In: Burgos Gil, J., Ebrahimi-Fard, K., Gangl, H. (eds) Periods in Quantum Field Theory and Arithmetic. ICMAT-MZV 2014. Springer Proceedings in Mathematics & Statistics, vol 314. Springer, Cham. https://doi.org/10.1007/978-3-030-37031-2_18
Download citation
DOI: https://doi.org/10.1007/978-3-030-37031-2_18
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-37030-5
Online ISBN: 978-3-030-37031-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)