References

  1. Akar M., Yüce S., Şahin S. On the dual hyperbolic numbers and the complex hyperbolic numbers. Journal of Computer Science and Computational Mathematics 2018, 8 (1), 1–6. doi:10.20967/jcscm.2018.01.001
  2. Alfsmann D. On families of 2n-dimensional hypercomplex algebras suitable for digital signal processing. In: 14th European Signal Processing Conf. (EUSIPCO 2006), Florence, Italy, September 4–8, 2006. doi:10.5281/ZENODO.52940
  3. Bergum G.E., Hoggat JR. V.E. Sums and products for recurring sequences. Fibonacci Quart. 1975, 13 (2), 115–120.
  4. Cheng H.H., Thompson S. Dual polynomials and complex dual numbers for analysis of spatial mechanisms. In: Proc. of ASME 24th Biennial Mechanisms Conference, Irvine, CA, August 19-22, 1996. doi:10.1115/96-DETC/MECH-1221
  5. Cihan A., Azak A.Z., Güngör M.A., Tosun M. A study on dual hyperbolic Fibonacci and Lucas numbers. Ann. Sci. Univ. “Ovidius” Constanta. Ser. Mat. 2019, 27 (1), 35–48. doi:10.2478/auom-2019-0002
  6. Cheng H.H., Thompson S. Singularity analysis of spatial mechanisms using dual polynomials and complex dual numbers. ASME. J. Mech. Des. 1999, 121 (2), 200–205. doi:10.1115/1.2829444
  7. Cockle J. On a new imaginary in algebra. Philosophical magazine. London-Dublin-Edinburgh 1849, 3 (34), 37–47. doi:10.1080/14786444908646169
  8. Cohen A., Shoham M. Principle of transference-an extension to hyper-dual numbers. Mech. Mach. Theory 2018, 125, 101–110. doi:10.1016/j.mechmachtheory.2017.12.007
  9. Fike J.A., Alonso J.J. The development of hyper-dual numbers for exact second- derivative calculations. In: The 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Orlando, Florida, January 4 - 7, 2011. doi:10.2514/6.2011-886
  10. Fike J.A., Alonso J.J. Automatic differentiation through the use of hyper-dual numbers for second derivatives. In: Forth S., Hovland P., Phipps E., Utke J., Walther A. (eds) Recent Advances in Algorithmic Differentiation. Lecture Notes in Computational Science and Engineering, 87, 163–173. Springer, Berlin, Heidelberg, 2012. doi:10.1007/978-3-642-30023-3_15
  11. Fjelstad P., Sorin Gal G. \(n\)-dimensional hyperbolic complex numbers. Adv. Appl. Clifford Algebr. 1998, 8 (1), 47–68. doi:10.1007/BF03041925
  12. Güngör M.A., Azak A.Z. Investigation of dual-complex Fibonacci, dual-complex Lucas numbers and their properties. Adv. Appl. Clifford Algebr. 2017, 27 (4), 3083–3096. doi:10.1007/s00006-017-0813-z
  13. Gürses N., Şentürk G.Y., Yüce S. A study on dual-generalized complex and hyperbolic-generalized complex numbers. Gazi University Journal of Science 2021, 34 (1), 180–194. doi:10.35378/gujs.653906
  14. Gürses N., Şentürk G.Y., Yüce S. A comprehensive survey of dual-generalized complex Fibonacci and Lucas numbers. Sigma J Eng Nat Sci 2022, 40 (1), 179–187. doi:10.14744/sigma.2022.00014
  15. Halici S. On Fibonacci quaternions. Adv. Appl. Clifford Algebr. 2012, 22 (2), 321–327. doi:10.1007/s00006-011-0317-1
  16. Halici S. On complex Fibonacci quaternions. Adv. Appl. Clifford Algebr. 2013, 23 (1), 105–112. doi:10.1007/s00006-012-0337-5
  17. Hamilton W.R. On quaternions; or on a new system of imaginaries in algebra. The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science (3rd Series) xxv-xxxvi, 1844–1850.
  18. Harkin A.A., Harkin J.B. Geometry of generalized complex numbers. Math. Mag. 2004, 77 (2), 118–129. doi:10.1080/0025570X.2004.11953236
  19. Horadam A.F. Complex Fibonacci numbers and Fibonacci quaternions. Amer. Math. Monthly 1963, 70 (3), 289–291. doi:10.2307/2313129
  20. Horadam A.F. Quaternion recurrence relations. Ulam Quarterly 1993, 2 (2), 23–33.
  21. Iyer M.R. Some results on Fibonacci quaternions. Fibonacci Quart. 1969, 7 (2), 201–210.
  22. Iyer M.R. A note on Fibonacci quaternions. Fibonacci Quart. 1969, 3 (7), 225–229.
  23. Kantor I., Solodovnikov A. Hypercomplex numbers: an elementary introduction to algebras. Springer-Verlag, New York, 1989.
  24. Majernik V. Multicomponent number systems. Acta Phys. Polon. A 1996, 90 (3), 491–498. doi:10.12693/APhysPolA.90.491
  25. Messelmi F. Dual-complex numbers and their holomorphic functions. doi:10.5281/ZENODO.22961
  26. Nurkan S.K, Güven I.A. Dual Fibonacci quaternions. Adv. Appl. Clifford Algebr. 2015, 25 (2), 403–414. doi:10.1007/s00006-014-0488-7
  27. Pennestrı̀ E., Stefanelli R. Linear algebra and numerical algorithms using dual numbers. Multibody Syst. Dyn. 2007, 18 (3), 323–344. doi:10.1007/s11044-007-9088-9
  28. Price G.B. An introduction to multicomplex spaces and functions. Monographs and textbooks in pure and applied mathematics, New-York. 1991. doi:10.1201/9781315137278
  29. Rochon D., Shapiro M. On algebraic properties of bicomplex and hyperbolic numbers. An. Univ. Oradea Fasc. Mat. 2004, 11, 71–110.
  30. Sobczyk G. The hyperbolic number plane. College Math. J. 1995, 26 (4), 268–280. doi:10.1080/07468342.1995.11973712
  31. Study E. Geometrie der dynamen: Die zusammensetzung von kräften und verwandte gegenstände der geometrie bearb. Leipzig, B.G. Teubner, 1903.
  32. Tan E., Ait-Amrane N. R., Gök I. Hyper-dual Horadam quaternions. Miskolc Math. Notes 2021, 22 (2), 903–913. doi:10.18514/MMN.2021.3747
  33. Toyoshima H. Computationally efficient bicomplex multipliers for digital signal processing. IEICE Trans Inf Syst. 1989, E81-D (2), 236–238.
  34. Yaglom I.M. Complex numbers in geometry. Academic Press, New York, 1968.
  35. Yaglom I.M. A simple non-Euclidean Geometry and its physical basis: An Elementary Account of Galilean Geometry and the Galilean Principle of Relativity. Heidelberg Science Library, 1979.