A new generalization of Lucas quaternions with finite operators

In this paper, we introduce a new family of Lucas quaternions by using finite operators. We call these quaternions as Lucas finite operator quaternions. We give some properties and identities of Lucas finite operator quaternions such as Binet-like formula, generating function, exponential generating function, Catalan's identity, Cassini's identity, d'Ocagne's identity and many binomial-sum identities. As an application, we generate Cassini's identity in another form by matrix representations.