Given a commutative ring with identity R, many different and interesting operations can be defined over the set HR of sequences of elements in R. These operations can also give HR 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 HR 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.

On the operations over sequences in rings and binomial type sequences / Barbero, Stefano; Cerruti, Umberto; Murru, Nadir. - In: RICERCHE DI MATEMATICA. - ISSN 1827-3491. - 67:(2018), pp. 911-927. [10.1007/s11587-018-0389-5]

On the operations over sequences in rings and binomial type sequences

Barbero, Stefano;Murru, Nadir
2018-01-01

Abstract

Given a commutative ring with identity R, many different and interesting operations can be defined over the set HR of sequences of elements in R. These operations can also give HR 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 HR 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.
2018
Barbero, Stefano; Cerruti, Umberto; Murru, Nadir
On the operations over sequences in rings and binomial type sequences / Barbero, Stefano; Cerruti, Umberto; Murru, Nadir. - In: RICERCHE DI MATEMATICA. - ISSN 1827-3491. - 67:(2018), pp. 911-927. [10.1007/s11587-018-0389-5]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/272189
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