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.
Similar content being viewed by others
References
Allouche, J.P., France, M.M.: Hadamard grade of power series. J. Number Theory 131(11), 2013–2022 (2011)
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)
Benhissi, A.: Ideal structure of Hurwitz series rings. Contrib. Algebra Geom. 48(1), 251–256 (2007)
Benhissi, A., Koja, F.: Basic properties of Hurwitz series rings. Ricerche Mat. 61(2), 255–273 (2012)
Charalambides, C.A.: Enumerative Combinatorics. Chapman and Hall, Boca Raton (2002)
Di Bucchianico, A., Loeb, D.: Sequences of binomial type with persistant roots. J. Math. Anal. Appl. 199(1), 39–58 (1996)
Di Bucchianico, A.: Probabilistic and Analytic Aspects of the Umbral Calculus. Centrum Wiskunde and Informatica (CWI), Amsterdam (1997)
Fill, J.A., Flajolet, P., Kapur, N.: Singularity analysis, Hadamard products, and tree recurrences. J. Comput. Appl. Math. 174, 271–313 (2005)
Ghanem, M.: Some properties of Hurwitz series ring. Int. Math. Forum 6(40), 1973–1981 (2007)
Goss, D.: Polynomials of binomial type and Lucas’ theorem. Proc. Am. Math. Soc. 144, 1897–1904 (2016)
Graham, R., Knuth, D., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science, 2nd edn. Addison-Wesley, Boston (1994)
Hadamard, J.: Theorem sur les series entieres. Acta Math. 22, 55–63 (1899)
Keigher, W.F., Pritchard, F.L.: Hurwitz series as formal functions. J. Pure Appl. Algebra 146, 291–304 (2000)
Kisil, V.V.: Polynomial sequences of binomial type path integrals. Ann. Comb. 6(1), 45–56 (2002)
Mihoubi, M.: Bell polynomials and binomial type sequences. Discrete Math. 308, 2450–2459 (2008)
Rota, G.C., Kahaner, D., Odlyzko, A.: Finite operator calculus. J. Math. Anal. Appl. 42, 685–760 (1973)
Schneider, J.: Polynomial sequences of binomial-type arising in graph theory. Electron. J. Comb. 21(1), 1.43 (2014)
The On-Line Encyclopedia of Integer Sequences. http://oeis.org/
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Barbero, S., Cerruti, U. & Murru, N. On the operations of sequences in rings and binomial type sequences. Ricerche mat 67, 911–927 (2018). https://doi.org/10.1007/s11587-018-0389-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11587-018-0389-5