On $3$-parameter generalized quaternions with higher order Leonardo numbers components

In this paper, a novel type of $3$-parameter generalized quaternions ( $3$ -PGQs) is introduced, constructed from higher order Leonardo numbers and referred to as the higher order Leonardo $3$-parameter generalized quaternions (shortly, higher order Leonardo $3$-PGQs). Several fundamental properties of these quaternions are examined, including their recurrence relations, a Binet-type formula, and both generating and exponential generating functions. In addition, the obtained identities show that the higher order Leonardo $3$-PGQs can be expressed in closed form in terms of Fibonacci numbers, Fibonacci generalized quaternions, and higher order Leonardo numbers.