Quaternion-Type Catalan Transforms of the ρ-Fibonacci and ρ-Lucas Numbers

Abstract

In this paper, we define a new sequence called the quaternion-type Catalan sequence and give generating function, exponential representation, quaternionic Catalan matrix and its some properties. Then, we introduce a quaternion-type Catalan transform, which is a generalization of the Catalan transform, and its inverse. Moreover, we apply the quaternion-type Catalan transforms to $\rho$ -Fibonacci and $\rho$ -Lucas sequences and derive some new sequences by using these quaternion-type Catalan transforms.