Representation of some special functions on transcendence basis
Vol. 129 No. 1B (2020), Original Article
Vol. 129 No. 1B (2020)

Representation of some special functions on transcendence basis

Original Article
https://doi.org/10.26459/hueuni-jns.v129i1B.5636
Published 2020-06-22
Bui Van Chien
University of Sciences, Hue University, 77 Nguyen Hue St., Hue, Vietnam
PDF

Keywords

Quasi-Shuffle Product
Special Functions
Harmonic Sum
Polyzetas
Polylogarithm Function.

How to Cite

1.
Chien BV. Representation of some special functions on transcendence basis. hueuni-jns [Internet]. 2020Jun.22 [cited 2024Nov.23];129(1B):79-86. Available from: http://222.255.146.83/index.php/hujos-ns/article/view/5636
ACM ACS APA ABNT Chicago Harvard IEEE MLA Turabian Vancouver

Download Citation

Endnote/Zotero/Mendeley (RIS) BibTeX
Crossref
Scopus
Google Scholar
Europe PMC

Abstract

The special functions such as multiple harmonic sums, polyzetas or multiple polylogarithm functions are compatible with quasi-shuffle algebras. By using transcendence bases of the quasi-shuffle algebras studied in the paper [4], we will express non-commutative generating series of these special functions and then identify on the local coordinates to reduce their polynomial relations or asymptotic expansions indexed by these bases.
https://doi.org/10.26459/hueuni-jns.v129i1B.5636
PDF

References

  1. Chien BV, Duchamp GHE, Minh VHN. Schützenberger's factorization on the (completed) Hopf algebra of q-stuffle product. JP J Algebra Number Theory Appl. 2013;30(2):191-215.
  2. Reutenauer C. Free Lie algebras, volume 7 of London Mathematical Society Monographs. Oxford: Oxford University Press; 1993.
  3. Minh HN, Petitot M, Hoeven JVD. Shuffle algebra and polylogarithms. Discrete Mathematics. 2000;225(1-3):217-230.
  4. Zagier D. Values of Zeta Functions and Their Applications. In: Joseph A, Mignot F, Murat F, Prum B, Rentschler R, editors. First European Congress of Mathematics Paris, July 6–10, 1992: Vol II: Invited Lectures (Part 2). Basel: Birkhäuser Basel; 1994. p. 497-512.
  5. Ree R. Lie elements and an algebra associated with shuffles. Ann of Math. 1958;68(2):210-220.
  6. Kuo-tsai C. Algebras of iterated path integrals and fundamental groups. Trans Amer Math Soc. 1971 ;156:359-379.
  7. Hoffman ME. The Algebra of Multiple Harmonic Series. Journal of Algebra. 1997;194(2):477-495.
  8. Radford DE. A natural ring basis for the shuffle algebra and an application to group schemes. Journal of Algebra. 1979;58(2):432-454.
  9. Chien BV, Duchamp GHE, Minh VHN. Structure of polyzetas and explicit representation on transcendence bases of shuffle and stuffle algebras. J Symbolic Comput. 2017;83:93-111.
  10. Chien BV. Développement asymptotique des sommes harmoniques [dissertation]. Paris: Laboratoire LIPN – Université Paris 13; 2016.
  11. Minh HN, Petitot M. Lyndon words, multiple polylogarithms and the Riemann function. Discrete Math. 2000;217(1-3):273-292.
Creative Commons License

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Copyright (c) 2020 Array