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.
Full version: pdf,
(Concerned with sequences
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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