Companion sequences associated to ther-Fibonacci sequence: algebraic andcombinatorial properties

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.

Bi-periodic $r$-Fibonacci sequence and bi-periodic $r$-Lucas sequence of type $s$

“…For more details on these sequences, we refer the reader to [1,4,6,8,12]. Each sequence in the family of companion sequences, the bi-periodic r-Lucas sequence of type s, satisfies the following linear recurrence relation.…”

“…n reduce to the r-Fibonacci numbers. Abbad et al [1] defined its family of companion sequences; the r-Lucas sequences of type s, for a positive integers r, s with 1 ≤ s ≤ r and real numbers x and y, by…”

“…The first one, is to introduce the parameters c and d in the expression of the recurrence sequences given by Yazlik et al in [12]. The second one, is to define a family of companion sequences as introduced in [1] for the bi-periodic case.…”

“…and nonzero real numbers a, b, c and d. Its linear recurrence relation is given by U(1) n = (ab + c + d)U ab + d. Its generating function is…”

ON 3-PARAMETER GENERALIZED QUATERNIONS WITH THE LEONARDO p-SEQUENCE

“…For any given integer and , the Fibonacci -sequence, and the Lucas -sequence are defined recursively by and with initial conditions and for , respectively, [3][4][5][6][7][8]. Very recently, for any given integer and , the Leonardo -sequence is defined recursively by the following non-homogeneous relation (1.1) with initial conditions for , [9].…”