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.
$r$-Fibonacci sequence, companion sequences, recurrence relation, convolution, hyper-$r$-Lucas polynomial, incomplete $r$-Lucas polynomial, $q$-analogues
ABBAD, SADJIA; BELBACHIR, HACENE; and BENZAGHOU, BENALI
"Companion sequences associated to the$r$-Fibonacci sequence: algebraic and combinatorial properties,"
Turkish Journal of Mathematics: Vol. 43:
3, Article 4.
Available at: https://journals.tubitak.gov.tr/math/vol43/iss3/4