Dual quaternion theory over HGC numbers

SAÇLI G. Y., GÜRSES N.

Journal of Discrete Mathematical Sciences and Cryptography, vol.27, no.1, pp.117-142, 2024 (ESCI, Scopus) identifier

  • Publication Type: Article / Article
  • Volume: 27 Issue: 1
  • Publication Date: 2024
  • Doi Number: 10.47974/jdmsc-1611
  • Journal Name: Journal of Discrete Mathematical Sciences and Cryptography
  • Journal Indexes: Emerging Sources Citation Index (ESCI), Scopus, Academic Search Premier, Aerospace Database, Communication Abstracts, INSPEC, Metadex, zbMATH, Civil Engineering Abstracts
  • Page Numbers: pp.117-142
  • Keywords: Dual quaternion, Fibonacci number, Hyperbolic-generalized complex number, Lucas number, Q-matrix
  • Istanbul Gelisim University Affiliated: Yes

Abstract

Knowing the applications of quaternions in various fields, such as robotics, navigation, computer visualization and animation, in this study, we give the theory of dual quaternions considering Hyperbolic-Generalized Complex (HGC) numbers as coefficients via generalized complex and hyperbolic numbers. We account for how HGC number theory can extend dual quaternions to HGC dual quaternions. Some related theoretical results with HGC Fibonacci/Lucas numbers are established, including their dual quaternions. Given HGC Fibonacci/Lucas numbers, their special matrix correspondences have been identified and these are carried out to HGC Fibonacci/Lucas dual quaternions. Furthermore, we provide a more accurate way to quickly calculate HGC Fibonacci numbers and associate this with HGC generalized Fibonacci numbers. For implementation, we produce an algorithm in Maple. Lastly, we put the theory into practice.