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

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Abstract

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.

Keywords

higher order Leonardo numbers / 3-parameter generalized quaternions / Binet-type formulas / generating functions

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