A New Family of k-Quasi Morgan-Voyce Sequences

Abstract

In this study, we define the kQuasi Morgan-Voyce and k-Quasi MorganVoyce-Lucas sequences, and some terms of these sequences are given. We introduce the closed-form formulas that give the terms of these sequences. Also, we examine the characteristic equation and properties of these sequences. Then, we find the relations between the terms of the k-Quasi Morgan-Voyce and kQuasi Morgan-Voyce-Lucas sequences. Also, we give these sequences generating functions, summation formulas, etc. We present special relations between the positive index term and the negative index term of these sequences. In addition, we obtain the Binet formulas in two different ways. The first method is the classic one. The second method is obtained with the help of the generating functions of the sequences. Moreover, we calculate the special identities of these sequences like Cassini and Catalan. Finally, we examine the relations of the k-Quasi MorganVoyce sequence with the Fibonacci, Bronze Fibonacci, Pell, Balancing, Jacobsthal, Mersenne, Oresme sequences and k-Quasi Morgan-VoyceLucas sequence with the Lucas, Bronze Lucas, Pell-Lucas, Balancing-Lucas, Jacobsthal-Lucas, Mersenne-Lucas, Oresme-Lucas sequences, respectively. In addition, for special k values, these sequences are associated with the sequences in OEIS.