Companion sequences associated to the $r$-Fibonacci sequence: algebraic and combinatorial properties

Authors : Sadjia ABBAD, Hacene BELBACHIR, Benali BENZAGHOU

Pages : 1095-1114

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Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :It is well known that the companion sequence of the Fibonacci sequence is Lucas`s sequence. For the generalized Fibonacci sequences, the companion sequence is not unique. Several authors proposed different definitions, and they are in a certain sense all good. Our purpose is to introduce a family of companion sequences for some generalized Fibonacci sequence: the $r$-Fibonacci sequence. We evaluate the generating functions and give some applications, and we exhibit convolution relations that generalize some known identities such as Cassini`s. Afterwards, we calculate the sums of their terms using matrix methods. Next, we propose a $q$-analogue and extend the definition to negative $n$s. Also, we define the incomplete associated sequences using a Euler--Seidel-like approach.
Keywords : r Fibonacci sequence, companion sequences, recurrence relation, convolution, hyper r Lucas polynomial, incomplete r Lucas polynomial, q analogues