On the operations of sequences in rings and binomial type sequences

Article
  • 16 Downloads

Abstract

Given a commutative ring with identity R, many different and interesting operations can be defined over the set \(H_R\) of sequences of elements in R. These operations can also give \(H_R\) the structure of a ring. We study some of these operations, focusing on the binomial convolution product and the operation induced by the composition of exponential generating functions. We provide new relations between these operations and their invertible elements. We also study automorphisms of the Hurwitz series ring, highlighting that some well-known transforms of sequences (such as the Stirling transform) are special cases of these automorphisms. Moreover, we introduce a novel isomorphism between \(H_R\) equipped with the componentwise sum and the set of the sequences starting with 1 equipped with the binomial convolution product. Finally, thanks to this isomorphism, we find a new method for characterizing and generating all the binomial type sequences.

Keywords

Binomial type sequence Binomial convolution product Hurwitz series ring 

Mathematics Subject Classification

11B99 11T06 13F25 

References

  1. 1.
    Allouche, J.P., France, M.M.: Hadamard grade of power series. J. Number Theory 131(11), 2013–2022 (2011)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Barbero, S., Cerruti, U., Murru, N.: Some combinatorial properties of the Hurwitz series ring, Ricerche di Matematica (to appear).  https://doi.org/10.1007/s11587-017-0336-x (2017)
  3. 3.
    Benhissi, A.: Ideal structure of Hurwitz series rings. Contrib. Algebra Geom. 48(1), 251–256 (2007)MathSciNetMATHGoogle Scholar
  4. 4.
    Benhissi, A., Koja, F.: Basic properties of Hurwitz series rings. Ricerche Mat. 61(2), 255–273 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Charalambides, C.A.: Enumerative Combinatorics. Chapman and Hall, Boca Raton (2002)MATHGoogle Scholar
  6. 6.
    Di Bucchianico, A., Loeb, D.: Sequences of binomial type with persistant roots. J. Math. Anal. Appl. 199(1), 39–58 (1996)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Di Bucchianico, A.: Probabilistic and Analytic Aspects of the Umbral Calculus. Centrum Wiskunde and Informatica (CWI), Amsterdam (1997)MATHGoogle Scholar
  8. 8.
    Fill, J.A., Flajolet, P., Kapur, N.: Singularity analysis, Hadamard products, and tree recurrences. J. Comput. Appl. Math. 174, 271–313 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ghanem, M.: Some properties of Hurwitz series ring. Int. Math. Forum 6(40), 1973–1981 (2007)MathSciNetMATHGoogle Scholar
  10. 10.
    Goss, D.: Polynomials of binomial type and Lucas’ theorem. Proc. Am. Math. Soc. 144, 1897–1904 (2016)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Graham, R., Knuth, D., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science, 2nd edn. Addison-Wesley, Boston (1994)MATHGoogle Scholar
  12. 12.
    Hadamard, J.: Theorem sur les series entieres. Acta Math. 22, 55–63 (1899)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Keigher, W.F., Pritchard, F.L.: Hurwitz series as formal functions. J. Pure Appl. Algebra 146, 291–304 (2000)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kisil, V.V.: Polynomial sequences of binomial type path integrals. Ann. Comb. 6(1), 45–56 (2002)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Mihoubi, M.: Bell polynomials and binomial type sequences. Discrete Math. 308, 2450–2459 (2008)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Rota, G.C., Kahaner, D., Odlyzko, A.: Finite operator calculus. J. Math. Anal. Appl. 42, 685–760 (1973)CrossRefMATHGoogle Scholar
  17. 17.
    Schneider, J.: Polynomial sequences of binomial-type arising in graph theory. Electron. J. Comb. 21(1), 1.43 (2014)MathSciNetGoogle Scholar
  18. 18.
    The On-Line Encyclopedia of Integer Sequences. http://oeis.org/

Copyright information

© Università degli Studi di Napoli "Federico II" 2018

Authors and Affiliations

  1. 1.Department of Mathematics G. PeanoUniversity of TurinTurinItaly

Personalised recommendations