q-Leonardo Split Quaternions and Some Applications
Ali Atasoy *
Keskin Vocational School, Kırıkkale University, Kırıkkale, T¨urkiye.
Faik Babadag
Department of Mathematics, Kırıkkale University, Kırıkkale, T¨urkiye.
*Author to whom correspondence should be addressed.
Abstract
Quantum calculus, using in various mathematical disciplines such as combinatorics and special functions, as well as in numerous applications including fractal analysis, multifractal measures, and entropy formulations of chaotic dynamical systems, has attracted significant scholarly attention in recent years. In the present study, we present a new class of Leonardo split quaternions whose components involve quantum integers into their components. We also present the fundamental properties and identities related to the q-Leonardo split quaternion, including Binet’s formula, exponential generating functions, binomial sums, as well as d’Ocagne’s,
Vajda’s, Catalan’s, and Cassini’s identities. Finally, we give different polar representation using Cayley Dickson’s notation applications for some q-Leonardo split quaternions.The applications can be converted into quantum integer forms under suitable conditions with similar considerations and give a deeper understanding of their algebraic and geometric interpretations, and transformations.
Keywords: q-Leonardo split quaternion, q-calculus, Leonardo number sequence