Octonions and hyperbolic octonions with the k-balancing and k-Lucas balancing numbers

In this paper, we introduce the k-balancing and k-Lucas balancing octonions and hyperbolic octonions. We give Binet type formulas of them and obtain some well-known identities like Cassini’s identity, d’Ocagne’s identity, Catalan identity, Vajda’s identity, generating functions, etc. for these octonions in closed form. Also we present some interesting properties of them. As a special case for \(k=1\) $k=1$ , the results immediately follow for balancing and Lucas-balancing octonions and hyperbolic octonions.