A note on hyperbolic (p,q)-Fibonacci quaternions

Abstract

In this paper, we introduce a new quaternion sequence called hyperbolic (p,q)-Fibonacci quaternions. This new quaternion sequence includes hyperbolic Fibonacci, hyperbolic k-Fibonacci, hyperbolic Pell, hyperbolic k-Pell, hyperbolic Jacobsthal, hyperbolic k-Jacobsthal quaternions. We give generating function and Binet formula for these quaternions. We also obtain some identities such as d'Ocagne's, Catalan's and Cassini's identities involving hyperbolic (p,q)-Fibonacci quaternions.