In this paper we study the action of the Binomial and Invert (interpolated) operators on the set of linear recurrent sequences. We prove that these operators preserve this set, and we find how they change the characteristic polynomials. We show that these operators, with the aid of two other elementary operators (essentially the left and right shifts), can transform any impulse sequence (a linear recurrent sequence starting from (0,0,...,0,1)) into any other impulse sequence, by two processes that we call construction and deconstruction. Finally, we give some applications to polynomial sequences and pyramidal numbers. We also find a new identity on Fibonacci numbers, and we prove that r–bonacci numbers are a Bell polynomial transform of the (r−1)–bonacci numbers.

### Tansforming recurrent sequences by using the Binomial and Invert operators

#### Abstract

In this paper we study the action of the Binomial and Invert (interpolated) operators on the set of linear recurrent sequences. We prove that these operators preserve this set, and we find how they change the characteristic polynomials. We show that these operators, with the aid of two other elementary operators (essentially the left and right shifts), can transform any impulse sequence (a linear recurrent sequence starting from (0,0,...,0,1)) into any other impulse sequence, by two processes that we call construction and deconstruction. Finally, we give some applications to polynomial sequences and pyramidal numbers. We also find a new identity on Fibonacci numbers, and we prove that r–bonacci numbers are a Bell polynomial transform of the (r−1)–bonacci numbers.
##### Scheda breve Scheda completa Scheda completa (DC)
2010
13
Article 10.7.7
1
16
http://arxiv.org/pdf/1105.2143.pdf
http://www.cs.uwaterloo.ca/journals/JIS/
bell polynomials; binomial operator; fibonacci numbers; impulse sequences; invert operator; pyramidal numbers; recurrent sequences
Stefano Barbero; Umberto Cerruti; Nadir Murru
File in questo prodotto:
File
20 - Transforming Recurrent Sequences by using Binomial and Invert Operators.pdf

Accesso aperto

Tipo di file: PDF EDITORIALE
Dimensione 185.94 kB
Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/2318/75553`