Skip to main content

The algebraic structure of linearly recursive sequences under hadamard product

Israel Journal of Mathematics volume 72, pages 118–132 (1990)Cite this article

Abstract

We describe the algebraic structure of linearly recursive sequences under the Hadamard (point-wise) product. We characterize the invertible elements and the zero divisors. Our methods use the Hopf-algebraic structure of this algebra and classical results on Hopf algebras. We show that our criterion for invertibility is effective if one knows a linearly recursive relation for a sequence and certain information about finitely-generated subgroups of the multiplicitive group of the field.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. M. F. Atyah and I. G. Macdonald,Introduction to Commutative Algebra, Addison-Wesley, Reading, 1969.

    Google Scholar 

  2. B. Benzaghou,Algèbres de Hadamard, Bull. Soc. Math. France98 (1970), 209–252.

    MATH  MathSciNet  Google Scholar 

  3. C. W. Curtis and I. Reiner,Representation Theory of Finite Groups and Associative Algebras, Wiley-Interscience, New York, 1962.

    MATH  Google Scholar 

  4. D. S. Passman,Infinite Group Rings, Marcel Dekker, New York, 1971.

    MATH  Google Scholar 

  5. B. Peterson and E. J. Taft,The Hopf algebra of linearly recursive sequences, Æquationes Math.20 (1980), 1–17.

    Article  MATH  MathSciNet  Google Scholar 

  6. A. J. van der Poorten,Some facts that should be better known, especially about rational functions, inNumber Theory and Applications (R. A. Mollin, ed.), Kluwer Acad. Publ., Dordrecht, 1989.

    Google Scholar 

  7. C. Reutenauer,Sur les éléments inversibles de l’algèbre de Hadamard des séries rationelles, Bull. Soc. Math. France110 (1982), 225–232.

    MATH  MathSciNet  Google Scholar 

  8. C. Ronse,Feedback Shift Registers, Springer-Verlag, Berlin, 1984.

    MATH  Google Scholar 

  9. M. Sweedler,Hopf Algebras, Benjamin, New York, 1969.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Chicago, 60680, Chicago, IL, USA

    Richard G. Larson

  2. Department of Mathematics, Rutgers University, 08903, New Brunswick, NJ, USA

    Earl J. Taft

Authors
  1. Richard G. Larson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Earl J. Taft
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Supported in part by NSF Grant DMS 870-1085.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Larson, R.G., Taft, E.J. The algebraic structure of linearly recursive sequences under hadamard product. Israel J. Math. 72, 118–132 (1990). https://doi.org/10.1007/BF02764615

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02764615

Keywords

  • Hopf Algebra
  • Algebraic Structure
  • Algebra Homomorphism
  • Zero Divisor
  • Hadamard Product