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Padovan, Perrin and Pell-Padovan Dual Quaternions

Year 2023, Volume: 15 Issue: 1, 125 - 144, 30.06.2023
https://doi.org/10.47000/tjmcs.999069

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.

References

  • 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.
  • Bilgici, G., Generalized order-$k$ Pell-Padovan-like numbers by matrix methods, Pure and Applied Mathematics Journal, 2(6) (2013), 174-178.
  • Cerda-Morales, G., Dual third order Jacobsthal quaternions, Proyecciones Journal of Mathematics, 37(4), (2018), 731-747.
  • Cerda-Morales, G., New identities for Padovan numbers, arXiv.org, (2019). https://arxiv.org/abs/1904.05492
  • Cerda-Morales, G., On a generalization for Tribonacci quaternions, Mediterr. J. Math., 14(6), Article number: 239 (2017), 12 pages.
  • Çimen, C. B., İpek, A., On Pell quaternions and Pell-Lucas quaternions, Advances in Applied Clifford Algebras, 26(1) (2016), 39-51.
  • Deveci, Ö., The Pell-Padovan sequences and the Jacobsthal-Padovan sequences in finite groups, Utilitas Mathematica, 98 (2015) 257-270.
  • Deveci, Ö., Shannon, A. G., Pell-Padovan-circulant sequences and their applications, Notes on Number Theory and Discrete Mathematics, 23 (2017) 100-114.
  • Deveci, Ö., Aküzüm, Y., Karaduman, E., The Pell-Padovan $p$-sequences and its applications, Utilitas Mathematica, 98 (2015), 327-347.
  • Dişkaya, O., Menken, H. On the $(s,t)$-Padovan and $(s,t)$-Perrin quaternions, J. Adv. Math. Stud., 12(2) (2019), 186-192.
  • 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.
  • Ercan, Z., Yüce, S., On properties of the dual quaternions, European Journal of Pure and Applied Mathematics, 4(2)(2011), 142–146.
  • Günay, H., Taşkara, N. Some properties of Padovan quaternion, Asian-European Journal of Mathematics, 12(06), Art. No. 2040017, (2019), 8 pages.
  • Halıcı, S., Karataş, A., On a generalization for Fibonacci quaternions, Chaos, Solitons & Fractals, 98 (2017), 178-182.
  • 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.
  • Horadam, A. F., Complex Fibonacci numbers and Fibonacci quaternions, Amer. Math. Monthly, 70 (1963), 289-291.
  • İşbilir, Z., Gürses, N., Pell-Padovan generalized quaternions, Notes on Number Theory and Discrete Mathematics, 27(1) (2021), 171-187.
  • Kalman, D., Generalized Fibonacci numbers by matrix methods, The Fibonacci Quarterly, 20(1) (1982), 73-76.
  • Kaygısız, K., Bozkurt, D., $k$-generalized order-$k$ Perrin number presentation by matrix method, Ars Combinatoria, 105, (2012), 95-101.
  • Khompungson, K., Rodjanadid, B., Sompong, S., Some matrices in terms of Perrin and Padovan sequences, Thai Journal of Mathematics, 17(3) (2019), 767-774.
  • Kızılateş, C., Catarino, P.M.M.C., Tuğlu, N., On the bicomplex generalized Tribonacci quaternions, Mathematics, 7(1)(2019), 80.
  • Lucas, E., Théorie des fonctions numériques simplement périodiques, Am. J. Math., 1(3) (1878), 197-240.
  • Majernik, V., Quaternion formulation of the Galilean space-time transformation, Acta Physica Slovaca, 56(1) (2006), 9-14.
  • 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.
  • Padovan, R., Dom Hans Van der Laan: Modern Primitive, Architectura & Natura Press, Amsterdam, (1994).
  • Padovan, R., Dom Hans Van der Laan and the plastic number, Nexus Network Journal, 4 (3) (2002), 181-193.
  • Perrin, R., Query 1484, J. Intermed. Math., 6 (1899), 76-77.
  • 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.
  • Shannon, A. G., Horadam, A. F. Some properties of third-order recurrence relations, The Fibonacci Quarterly, 10(2) (1972), 135-146.
  • 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.
  • 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.
  • 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.
  • Sloane, N.J.A., The on-line encyclopedia of integer sequences, (1964). Available online at: \url{http://oeis.org/}
  • Sokhuma, K., Matrices formula for Padovan and Perrin sequences, Applied Mathematical Sciences, 7 (142) (2013), 7093-7096.
  • Sokhuma, K., Padovan $Q$-matrix and the generalized relations, Applied Mathematical Sciences, 7(56) (2013), 2777-2780.
  • 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.
  • Soykan, Y., Generalized Pell-Padovan numbers, Asian Journal of Advanced Research and Reports, 11 (2) (2020) 8-28.
  • Soykan, Y., Summing formulas for generalized Tribonacci numbers, Universal Journal of Mathematics and Applications, 3(1) (2020), 1-11.
  • Stewart, I., Math Hysteria: Fun and Games with Mathematics, Oxford University Press, New York, (2004).
  • Stewart, I., Tales of a neglected number, Scientific American, 274 (1996), 102-103.
  • Szynal-Liana, A., Wloch, I., A note on Jacobsthal quaternions, Advances in Applied Clifford Algebras, 26 (19) (2016), 441-447.
  • Taşcı, D., Padovan and Pell-Padovan quaternions, Journal of Science and Arts, 42(1) (2018), 125-132.
  • Waddill, M. E., Sacks, L., Another generalized Fibonacci sequence, The Fibonacci Quarterly, 5 (3) (1967), 209-222.
  • Yılmaz, F., Bozkurt, D., Some properties of Padovan sequence by matrix method, Ars Combinatoria, 104, (2012), 149-160.
  • Yılmaz, N., The matrix representations of Padovan and Perrin numbers, Selçuk University, Graduate School of Natural and Applied Sciences, PhD Thesis, Konya, (2015).
  • Yılmaz, N., Taşkara, N., Binomial transforms of the Padovan and Perrin matrix sequences, Abstr. Appl. Anal., {2013, 497418, (2013), 7 pages.
  • Yılmaz, N., Taşkara, N., Matrix sequences in terms of Padovan and Perrin numbers, J. App. Math., 2013, Article ID: 941673 (2013), 7 pages.
  • Yılmaz, N., Taşkara, N., On the negatively subscripted Padovan and Perrin matrix sequences, Communications in Mathematics and Applications, 5 (2) (2014), 59-72.
  • Yüce, S., Aydın, F. T., Generalized dual Fibonacci quaternions, Applied Mathematics E-Notes, 16 (309) (2016), 276-289.
Year 2023, Volume: 15 Issue: 1, 125 - 144, 30.06.2023
https://doi.org/10.47000/tjmcs.999069

Abstract

References

  • 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.
  • Bilgici, G., Generalized order-$k$ Pell-Padovan-like numbers by matrix methods, Pure and Applied Mathematics Journal, 2(6) (2013), 174-178.
  • Cerda-Morales, G., Dual third order Jacobsthal quaternions, Proyecciones Journal of Mathematics, 37(4), (2018), 731-747.
  • Cerda-Morales, G., New identities for Padovan numbers, arXiv.org, (2019). https://arxiv.org/abs/1904.05492
  • Cerda-Morales, G., On a generalization for Tribonacci quaternions, Mediterr. J. Math., 14(6), Article number: 239 (2017), 12 pages.
  • Çimen, C. B., İpek, A., On Pell quaternions and Pell-Lucas quaternions, Advances in Applied Clifford Algebras, 26(1) (2016), 39-51.
  • Deveci, Ö., The Pell-Padovan sequences and the Jacobsthal-Padovan sequences in finite groups, Utilitas Mathematica, 98 (2015) 257-270.
  • Deveci, Ö., Shannon, A. G., Pell-Padovan-circulant sequences and their applications, Notes on Number Theory and Discrete Mathematics, 23 (2017) 100-114.
  • Deveci, Ö., Aküzüm, Y., Karaduman, E., The Pell-Padovan $p$-sequences and its applications, Utilitas Mathematica, 98 (2015), 327-347.
  • Dişkaya, O., Menken, H. On the $(s,t)$-Padovan and $(s,t)$-Perrin quaternions, J. Adv. Math. Stud., 12(2) (2019), 186-192.
  • 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.
  • Ercan, Z., Yüce, S., On properties of the dual quaternions, European Journal of Pure and Applied Mathematics, 4(2)(2011), 142–146.
  • Günay, H., Taşkara, N. Some properties of Padovan quaternion, Asian-European Journal of Mathematics, 12(06), Art. No. 2040017, (2019), 8 pages.
  • Halıcı, S., Karataş, A., On a generalization for Fibonacci quaternions, Chaos, Solitons & Fractals, 98 (2017), 178-182.
  • 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.
  • Horadam, A. F., Complex Fibonacci numbers and Fibonacci quaternions, Amer. Math. Monthly, 70 (1963), 289-291.
  • İşbilir, Z., Gürses, N., Pell-Padovan generalized quaternions, Notes on Number Theory and Discrete Mathematics, 27(1) (2021), 171-187.
  • Kalman, D., Generalized Fibonacci numbers by matrix methods, The Fibonacci Quarterly, 20(1) (1982), 73-76.
  • Kaygısız, K., Bozkurt, D., $k$-generalized order-$k$ Perrin number presentation by matrix method, Ars Combinatoria, 105, (2012), 95-101.
  • Khompungson, K., Rodjanadid, B., Sompong, S., Some matrices in terms of Perrin and Padovan sequences, Thai Journal of Mathematics, 17(3) (2019), 767-774.
  • Kızılateş, C., Catarino, P.M.M.C., Tuğlu, N., On the bicomplex generalized Tribonacci quaternions, Mathematics, 7(1)(2019), 80.
  • Lucas, E., Théorie des fonctions numériques simplement périodiques, Am. J. Math., 1(3) (1878), 197-240.
  • Majernik, V., Quaternion formulation of the Galilean space-time transformation, Acta Physica Slovaca, 56(1) (2006), 9-14.
  • 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.
  • Padovan, R., Dom Hans Van der Laan: Modern Primitive, Architectura & Natura Press, Amsterdam, (1994).
  • Padovan, R., Dom Hans Van der Laan and the plastic number, Nexus Network Journal, 4 (3) (2002), 181-193.
  • Perrin, R., Query 1484, J. Intermed. Math., 6 (1899), 76-77.
  • 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.
  • Shannon, A. G., Horadam, A. F. Some properties of third-order recurrence relations, The Fibonacci Quarterly, 10(2) (1972), 135-146.
  • 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.
  • 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.
  • 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.
  • Sloane, N.J.A., The on-line encyclopedia of integer sequences, (1964). Available online at: \url{http://oeis.org/}
  • Sokhuma, K., Matrices formula for Padovan and Perrin sequences, Applied Mathematical Sciences, 7 (142) (2013), 7093-7096.
  • Sokhuma, K., Padovan $Q$-matrix and the generalized relations, Applied Mathematical Sciences, 7(56) (2013), 2777-2780.
  • 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.
  • Soykan, Y., Generalized Pell-Padovan numbers, Asian Journal of Advanced Research and Reports, 11 (2) (2020) 8-28.
  • Soykan, Y., Summing formulas for generalized Tribonacci numbers, Universal Journal of Mathematics and Applications, 3(1) (2020), 1-11.
  • Stewart, I., Math Hysteria: Fun and Games with Mathematics, Oxford University Press, New York, (2004).
  • Stewart, I., Tales of a neglected number, Scientific American, 274 (1996), 102-103.
  • Szynal-Liana, A., Wloch, I., A note on Jacobsthal quaternions, Advances in Applied Clifford Algebras, 26 (19) (2016), 441-447.
  • Taşcı, D., Padovan and Pell-Padovan quaternions, Journal of Science and Arts, 42(1) (2018), 125-132.
  • Waddill, M. E., Sacks, L., Another generalized Fibonacci sequence, The Fibonacci Quarterly, 5 (3) (1967), 209-222.
  • Yılmaz, F., Bozkurt, D., Some properties of Padovan sequence by matrix method, Ars Combinatoria, 104, (2012), 149-160.
  • Yılmaz, N., The matrix representations of Padovan and Perrin numbers, Selçuk University, Graduate School of Natural and Applied Sciences, PhD Thesis, Konya, (2015).
  • Yılmaz, N., Taşkara, N., Binomial transforms of the Padovan and Perrin matrix sequences, Abstr. Appl. Anal., {2013, 497418, (2013), 7 pages.
  • Yılmaz, N., Taşkara, N., Matrix sequences in terms of Padovan and Perrin numbers, J. App. Math., 2013, Article ID: 941673 (2013), 7 pages.
  • Yılmaz, N., Taşkara, N., On the negatively subscripted Padovan and Perrin matrix sequences, Communications in Mathematics and Applications, 5 (2) (2014), 59-72.
  • Yüce, S., Aydın, F. T., Generalized dual Fibonacci quaternions, Applied Mathematics E-Notes, 16 (309) (2016), 276-289.
There are 49 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Zehra İşbilir 0000-0001-5414-5887

Nurten Gürses 0000-0001-8407-854X

Publication Date June 30, 2023
Published in Issue Year 2023 Volume: 15 Issue: 1

Cite

APA İşbilir, Z., & Gürses, N. (2023). Padovan, Perrin and Pell-Padovan Dual Quaternions. Turkish Journal of Mathematics and Computer Science, 15(1), 125-144. https://doi.org/10.47000/tjmcs.999069
AMA İşbilir Z, Gürses N. Padovan, Perrin and Pell-Padovan Dual Quaternions. TJMCS. June 2023;15(1):125-144. doi:10.47000/tjmcs.999069
Chicago İşbilir, Zehra, and Nurten Gürses. “Padovan, Perrin and Pell-Padovan Dual Quaternions”. Turkish Journal of Mathematics and Computer Science 15, no. 1 (June 2023): 125-44. https://doi.org/10.47000/tjmcs.999069.
EndNote İşbilir Z, Gürses N (June 1, 2023) Padovan, Perrin and Pell-Padovan Dual Quaternions. Turkish Journal of Mathematics and Computer Science 15 1 125–144.
IEEE Z. İşbilir and N. Gürses, “Padovan, Perrin and Pell-Padovan Dual Quaternions”, TJMCS, vol. 15, no. 1, pp. 125–144, 2023, doi: 10.47000/tjmcs.999069.
ISNAD İşbilir, Zehra - Gürses, Nurten. “Padovan, Perrin and Pell-Padovan Dual Quaternions”. Turkish Journal of Mathematics and Computer Science 15/1 (June 2023), 125-144. https://doi.org/10.47000/tjmcs.999069.
JAMA İşbilir Z, Gürses N. Padovan, Perrin and Pell-Padovan Dual Quaternions. TJMCS. 2023;15:125–144.
MLA İşbilir, Zehra and Nurten Gürses. “Padovan, Perrin and Pell-Padovan Dual Quaternions”. Turkish Journal of Mathematics and Computer Science, vol. 15, no. 1, 2023, pp. 125-44, doi:10.47000/tjmcs.999069.
Vancouver İşbilir Z, Gürses N. Padovan, Perrin and Pell-Padovan Dual Quaternions. TJMCS. 2023;15(1):125-44.