Geodesic
Transforming recurrent sequences by using the binomial and invert operators
Journal of integer sequences, Tome 13 (2010) no. 7
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 determine 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 ($0, \dots , 0, 1$) ) 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.
Classification : 11B37, 11B39
Keywords: Bell polynomials, binomial operator, Fibonacci numbers, impulse sequences, invert operator, pyramidal numbers, recurrent sequences 
@article{JIS_2010__13_7_a0,
     author = {Barbero,  Stefano and Cerruti,  Umberto and Murru,  Nadir},
     title = {Transforming recurrent sequences by using the binomial and invert operators},
     journal = {Journal of integer sequences},
     year = {2010},
     volume = {13},
     number = {7},
     zbl = {1207.11022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2010__13_7_a0/}
}
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Barbero,  Stefano; Cerruti,  Umberto; Murru,  Nadir. Transforming recurrent sequences by using the binomial and invert operators. Journal of integer sequences, Tome 13 (2010) no. 7. http://geodesic.mathdoc.fr/item/JIS_2010__13_7_a0/