Newton Series, Coinductively

  • Henning Basold
  • Helle Hvid Hansen
  • Jean-Éric Pin
  • Jan Rutten
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9399)


We present a comparative study of four product operators on weighted languages: (i) the convolution, (ii) the shuffle, (iii) the infiltration, and (iv) the Hadamard product. Exploiting the fact that the set of weighted languages is a final coalgebra, we use coinduction to prove that an operator of the classical difference calculus, the Newton transform, generalises (from infinite sequences) to weighted languages. We show that the Newton transform is an isomorphism of rings that transforms the Hadamard product of two weighted languages into their infiltration product, and we develop various representations for the Newton transform of a language, together with concrete calculation rules for computing them.


Product Operator Convolution Product Closed Formula Hadamard Product Binomial Coefficient 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We thank the anonymous referees for their constructive comments.


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Henning Basold
    • 1
  • Helle Hvid Hansen
    • 2
  • Jean-Éric Pin
    • 3
  • Jan Rutten
    • 1
    • 4
  1. 1.Radboud University NijmegenNijmegenNetherlands
  2. 2.Delft University of TechnologyDelftNetherlands
  3. 3.LIAFA Université Paris VII and CNRSParisFrance
  4. 4.CWI AmsterdamAmsterdamNetherlands

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