Journal of Integer Sequences, Vol. 13 (2010), Article 10.7.7

Transforming Recurrent Sequences by Using the Binomial and Invert Operators

Stefano Barbero, Umberto Cerruti, and Nadir Murru
Department of Mathematics
University of Turin
via Carlo Alberto 8/10


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, ... , 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.

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(Concerned with sequences A000045 A000073 A000078 A000217 A000290 A000292 A000326 A000384 A000566 A000567 A001106 A001107.)

Received September 14 2009; revised version received February 15 2010; March 9 2010; June 29 2010; July 10 2010. Published in Journal of Integer Sequences, July 16 2010.

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