aequationes mathematicae

, Volume 20, Issue 1, pp 1–17 | Cite as

The hopf algebra of linearly recursive sequences

  • Brian Peterson
  • Earl J. Taft
Survey Paper

Abstract

We explain how the space of linearly recursive sequences over a field can be considered as a Hopf algebra. The algebra structure is that of divided-power sequences, so we concentrate on the perhaps lesser-known coalgebra (diagonalization) structure. Such a sequence satisfies a minimal recursive relation, whose solution space is the subcoalgebra generated by the sequence. We discuss possible bases for the solution space from the point of view of diagonalization. In particular, we give an algorithm for diagonalizing a sequence in terms of the basis of the coalgebra it generates formed by its images under the difference-operator shift. The computation involves inverting the Hankel matrix of the sequence. We stress the classical connection (say over the real or complex numbers) with formal power series and the theory of linear homogeneous ordinary differential equations. It is hoped that this exposition will encourage the use of Hopf algebraic ideas in the study of certain combinatorial areas of mathematics.

AMS (1970) subject classification

primary 16A24 

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References

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    Hochschild, G.,The structure of Lie graups. Holden-Day, San Francisco, 1965.Google Scholar
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    Markushevich, A. I.,Recursion sequences (English translation). Mir Publishers, Moscow, 1975.Google Scholar
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    Taft, E. J.,Reflexivity of algebras and coalgebras. Amer. J. Math.94 (1972), 1111–1130.Google Scholar

Copyright information

© Birkhäuser Verlag 1980

Authors and Affiliations

  • Brian Peterson
    • 1
    • 2
  • Earl J. Taft
    • 1
    • 2
  1. 1.Department of MathematicsSan Jose State UniversitySan JoseU.S.A.
  2. 2.Department of MathematicsRutgers UniversityNew BrunswickU.S.A.

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