Evaluating Generating Functions for Periodic Multiple Polylogarithms via Rational Chen–Fliess Series

  • Kurusch Ebrahimi-Fard
  • W. Steven GrayEmail author
  • Dominique Manchon
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 314)


The goal of the paper is to give a systematic way to numerically evaluate the generating function of a periodic multiple polylogarithm using a Chen–Fliess series with a rational generating series. The idea is to realize the corresponding Chen–Fliess series as a bilinear dynamical system. A standard form for such a realization is given. The method is also generalized to the case where the multiple polylogarithm has non-periodic components. This allows one, for instance, to numerically validate the Hoffman conjecture. Finally, a setting in terms of dendriform algebras is provided.


Chen–Fliess series Dendriform algebra Hoffman conjecture Multiple polylogarithms Rational formal power series 



The second author was supported by grant SEV-2011-0087 from the Severo Ochoa Excellence Program at the Instituto de Ciencias Matemáticas in Madrid, Spain. This research was also supported by a grant from the BBVA Foundation.


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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Kurusch Ebrahimi-Fard
    • 1
  • W. Steven Gray
    • 2
    Email author
  • Dominique Manchon
    • 3
  1. 1.Department of Mathematical SciencesNorwegian University of Science and Technology (NTNU)TrondheimNorway
  2. 2.Old Dominion UniversityNorfolkUSA
  3. 3.LMBPCNRS - Université Clermont-AuvergneAubière CedexFrance

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