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

Abstract

It is well known that the companion sequence of the Fibonacci sequence is Lucas’s sequence. For thegeneralized 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 generalizedFibonacci sequence: the r -Fibonacci sequence. We evaluate the generating functions and give some applications, andwe exhibit convolution relations that generalize some known identities such as Cassini’s. Afterwards, we calculate thesums of their terms using matrix methods. Next, we propose a q -analogue and extend the definition to negative ns.Also, we define the incomplete associated sequences using a Euler–Seidel-like approach.