Representation of some special functions on transcendence basis
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 2020Oct.1];129(1B):79-86. Available from: http://jos.hueuni.edu.vn/index.php/HUJOS-NS/article/view/5636

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