On Split k-Fibonacci and k-Lucas Quaternions

Abstract

In this paper, we introduce the split k-Fibonacci and k-Lucas quaternions. We obtain the Binet formulas, generating functions and exponential generating functions of these quaternions. Moreover, we give the Catalan, Cassini and d'Ocagne identities for the split k-Fibonacci and k-Lucas quaternions.

Split Jacobsthal and Jacobsthal-Lucas Quaternions

“…From the Binet's formula (10), the proof of the identity (11) is completed. (12): From the Binet's formulas (9) and (10), we have…”

On generalized bicomplex k-Fibonacci numbers

“…Moreover, Ramirez [12] defined and studied k-Fibonacci and k-Lucas quaternions. Thereafter, Polatli et al [11] defined split k-Fibonacci and k-Lucas quaternions. Furthermore, Bilgici et al [3] introduced k-Fibonacci and k-Lucas generalized quaternions.…”

On metallic ratio in

“…The author investigated the generating functions and Binet's formulas for these quaternions and he derived some sums formulas and identities such as Cassini's identity. In 2016, Polatli et al introduced the split k ‐Fibonacci and k ‐Lucas quaternions. They obtained the Binet's formulas, generating functions, and exponential generating functions of these quaternions, and they gave the Catalan, Cassini, and d'Ocagne identities for the split k ‐Fibonacci and k ‐Lucas quaternions.…”