Padovan, Perrin and Pell-Padovan Dual Quaternions

Abstract

In this present study, we intend to determine the Padovan, Perrin and Pell-Padovan dual quaternions with nonnegative and negative subscripts. In line with this purpose, we construct some new properties such as; special determinant equalities, new recurrence relations, matrix formulas, Binet-like formulas, generating functions, exponential generating functions, summation formulas, and binomial properties for these special dual quaternions.

Keywords

• Padovan numbers
• Perrin numbers
• Pell-Padovan numbers
• dual quaternions.

References

1. Atanassov, K., Dimitrov, D., Shannon, A., A remark on $\psi$-function and Pell-Padovan's sequence, Notes on Number Theory and Discrete Mathematics, 15 (2009) 1-44.
2. Bilgici, G., Generalized order-$k$ Pell-Padovan-like numbers by matrix methods, Pure and Applied Mathematics Journal, 2(6) (2013), 174-178.
3. Cerda-Morales, G., Dual third order Jacobsthal quaternions, Proyecciones Journal of Mathematics, 37(4), (2018), 731-747.
4. Cerda-Morales, G., New identities for Padovan numbers, arXiv.org, (2019). https://arxiv.org/abs/1904.05492
5. Cerda-Morales, G., On a generalization for Tribonacci quaternions, Mediterr. J. Math., 14(6), Article number: 239 (2017), 12 pages.
6. Çimen, C. B., İpek, A., On Pell quaternions and Pell-Lucas quaternions, Advances in Applied Clifford Algebras, 26(1) (2016), 39-51.
7. Deveci, Ö., The Pell-Padovan sequences and the Jacobsthal-Padovan sequences in finite groups, Utilitas Mathematica, 98 (2015) 257-270.
8. Deveci, Ö., Shannon, A. G., Pell-Padovan-circulant sequences and their applications, Notes on Number Theory and Discrete Mathematics, 23 (2017) 100-114.

9. Deveci, Ö., Aküzüm, Y., Karaduman, E., The Pell-Padovan $p$-sequences and its applications, Utilitas Mathematica, 98 (2015), 327-347.
10. Dişkaya, O., Menken, H. On the $(s,t)$-Padovan and $(s,t)$-Perrin quaternions, J. Adv. Math. Stud., 12(2) (2019), 186-192.
11. Dişkaya, O., Menken, H., On the split $(s,t)$-Padovan and $(s,t)$-Perrin quaternions, International Journal of Applied Mathematics and Informatics, 13 (2019), 25-28.
12. Ercan, Z., Yüce, S., On properties of the dual quaternions, European Journal of Pure and Applied Mathematics, 4(2)(2011), 142–146.
13. Günay, H., Taşkara, N. Some properties of Padovan quaternion, Asian-European Journal of Mathematics, 12(06), Art. No. 2040017, (2019), 8 pages.
14. Halıcı, S., Karataş, A., On a generalization for Fibonacci quaternions, Chaos, Solitons & Fractals, 98 (2017), 178-182.
15. Hamilton, W. R., XI. On quaternions; or on a new system of imaginaries in algebra, The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, 33(219) (1848), 58-60.
16. Horadam, A. F., Complex Fibonacci numbers and Fibonacci quaternions, Amer. Math. Monthly, 70 (1963), 289-291.
17. İşbilir, Z., Gürses, N., Pell-Padovan generalized quaternions, Notes on Number Theory and Discrete Mathematics, 27(1) (2021), 171-187.
18. Kalman, D., Generalized Fibonacci numbers by matrix methods, The Fibonacci Quarterly, 20(1) (1982), 73-76.
19. Kaygısız, K., Bozkurt, D., $k$-generalized order-$k$ Perrin number presentation by matrix method, Ars Combinatoria, 105, (2012), 95-101.
20. Khompungson, K., Rodjanadid, B., Sompong, S., Some matrices in terms of Perrin and Padovan sequences, Thai Journal of Mathematics, 17(3) (2019), 767-774.
21. Kızılateş, C., Catarino, P.M.M.C., Tuğlu, N., On the bicomplex generalized Tribonacci quaternions, Mathematics, 7(1)(2019), 80.
22. Lucas, E., Théorie des fonctions numériques simplement périodiques, Am. J. Math., 1(3) (1878), 197-240.
23. Majernik, V., Quaternion formulation of the Galilean space-time transformation, Acta Physica Slovaca, 56(1) (2006), 9-14.
24. Mangueira, M.C. dos S., Vieira, R.P.M., Alves, F.R.V., Catarino, P.M.M.C., A generalizaçao da forma matricial da sequência de Perrin, Revista Sergipana de Matemática e Educação Matemática, 5 (1) (2020), 384-392.
25. Padovan, R., Dom Hans Van der Laan: Modern Primitive, Architectura & Natura Press, Amsterdam, (1994).
26. Padovan, R., Dom Hans Van der Laan and the plastic number, Nexus Network Journal, 4 (3) (2002), 181-193.
27. Perrin, R., Query 1484, J. Intermed. Math., 6 (1899), 76-77.
28. Seenukul, P., Netmanee, S., Panyakhun, T., Auiseekaen, R., Muangchan, S.-A., Matrices which have similar properties to Padovan $Q$-matrix and its generalized relations, SNRU Journal of Science and Technology, 7(2) (2015), 90-94.
29. Shannon, A. G., Horadam, A. F. Some properties of third-order recurrence relations, The Fibonacci Quarterly, 10(2) (1972), 135-146.
30. Shannon, A.G., Anderson, P.G., Horadam, A.F., Properties of Cordonnier, Perrin and Van der Laan numbers, International Journal of Mathematical Education in Science and Technology, 37(7) (2006), 825-831.
31. Shannon, A. G., Horadam, A. F., Anderson, P. G., The auxiliary equation associated with the plastic number, Notes on Number Theory and Discrete Mathematics, 12(1) (2006), 1-12.
32. Shannon, A. G., Wong, C. K., Some properties of generalized third order Pell numbers, Notes on Number Theory and Discrete Mathematics, 14(4) (2008), 16-24.
33. Sloane, N.J.A., The on-line encyclopedia of integer sequences, (1964). Available online at: \url{http://oeis.org/}
34. Sokhuma, K., Matrices formula for Padovan and Perrin sequences, Applied Mathematical Sciences, 7 (142) (2013), 7093-7096.
35. Sokhuma, K., Padovan $Q$-matrix and the generalized relations, Applied Mathematical Sciences, 7(56) (2013), 2777-2780.
36. Sompong, S., Wora-Ngon, N., Piranan, A., Wongkaentow, N., Some matrices with Padovan $Q$-matrix property, Proceedings of the 13-th IMT-GT International Conference on Mathematics, Statistics and their Applications (ICMSA2017), $4-7$ December 2017, Kedah, Malaysia, AIP Publishing LLC., 1905, 1, 030035, (2017), 6 pages.
37. Soykan, Y., Generalized Pell-Padovan numbers, Asian Journal of Advanced Research and Reports, 11 (2) (2020) 8-28.
38. Soykan, Y., Summing formulas for generalized Tribonacci numbers, Universal Journal of Mathematics and Applications, 3(1) (2020), 1-11.
39. Stewart, I., Math Hysteria: Fun and Games with Mathematics, Oxford University Press, New York, (2004).
40. Stewart, I., Tales of a neglected number, Scientific American, 274 (1996), 102-103.
41. Szynal-Liana, A., Wloch, I., A note on Jacobsthal quaternions, Advances in Applied Clifford Algebras, 26 (19) (2016), 441-447.
42. Taşcı, D., Padovan and Pell-Padovan quaternions, Journal of Science and Arts, 42(1) (2018), 125-132.
43. Waddill, M. E., Sacks, L., Another generalized Fibonacci sequence, The Fibonacci Quarterly, 5 (3) (1967), 209-222.
44. Yılmaz, F., Bozkurt, D., Some properties of Padovan sequence by matrix method, Ars Combinatoria, 104, (2012), 149-160.
45. Yılmaz, N., The matrix representations of Padovan and Perrin numbers, Selçuk University, Graduate School of Natural and Applied Sciences, PhD Thesis, Konya, (2015).
46. Yılmaz, N., Taşkara, N., Binomial transforms of the Padovan and Perrin matrix sequences, Abstr. Appl. Anal., {2013, 497418, (2013), 7 pages.
47. Yılmaz, N., Taşkara, N., Matrix sequences in terms of Padovan and Perrin numbers, J. App. Math., 2013, Article ID: 941673 (2013), 7 pages.
48. Yılmaz, N., Taşkara, N., On the negatively subscripted Padovan and Perrin matrix sequences, Communications in Mathematics and Applications, 5 (2) (2014), 59-72.
49. Yüce, S., Aydın, F. T., Generalized dual Fibonacci quaternions, Applied Mathematics E-Notes, 16 (309) (2016), 276-289.