Pell–Padovan generalized quaternions

Zehra İşbilir and Nurten Gürses
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 1, Pages 171—187
DOI: 10.7546/nntdm.2021.27.1.171-187
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Authors and affiliations

Zehra İşbilir
Duzce University, Faculty of Arts and Sciences,
Department of Mathematics, 81620, Duzce, Turkey

Nurten Gürses
Yildiz Technical University, Faculty of Arts and Sciences,
Department of Mathematics, 34220, Istanbul, Turkey


The aim of this article is to introduce Pell–Padovan generalized quaternions. It also derives new properties associated with these and takes into account negative indices. Additionally, it presents generating function, Binet-like formula, Simson formula, matrix representations, and several summation properties.


  • Pell–Padovan numbers
  • Generalized quaternions
  • Generating function
  • Binet-like formula
  • Simson formula

2010 Mathematics Subject Classification

  • 11B37
  • 11C20
  • 11K31
  • 11R52


  1. Bicknell, N. (1975). A primer on the Pell sequence and related sequences. The Fibonacci Quarterly, 13(4), 345—349.
  2. Bilgici, G. (2013). Generalized order–k Pell–Padovan-like numbers by matrix methods. Pure and Applied Mathematics Journal, 2(6), 174–178.
  3. Cerda-Morales, G. (2017). On a generalization for Tribonacci quaternions. Mediterranean Journal of Mathematics, 14(239), 12 pages.
  4. Cerda-Morales, G. (2019). New identities for Padovan numbers., Available online at:
  5. Cockle, J. (1849). On systems of algebra involving more than one imaginary, and on equations of the fifth degree. Philosophical Magazine, 35(238), 434–437.
  6. Daşdemir, A. (2011). On the Pell, Pell–Lucas and modified Pell numbers by matrix method. Applied Mathematical Sciences, 5(64), 3173–3181.
  7. Deveci, Ö. (2015). The Pell–Padovan sequences and the Jacobsthal–Padovan sequences in finite groups. Utilitas Mathematica, 98, 257–270.
  8. Deveci, O., Aküzüm, Y. & Karaduman, E. (2015). The Pell–Padovan p-sequences and its applications. Utilitas Mathematica, 98, 327–347.
  9. Dişkaya, O., & Menken, H. (2019). On the (s, t)-Padovan and (s, t)-Perrin quaternions. Journal of Advanced Mathematical Studies, 12(2), 186–192.
  10. Dişkaya, O., & Menken, H. (2019). On the split (s, t)-Padovan and (s, t)-Perrin quaternions. International Journal Of Applied Mathematıcs and Informatics, 13, 25–28.
  11. Dunlap, R. A. (1997). The Golden Ratio and Fibonacci Numbers. World Scientific,
  12. Ercolano, J. (1979). Matrix generators of Pell sequence. The Fibonacci Quarterly, 17(1), 71–77.
  13. Gunay, H., & Taşkara, N. (2019). Some properties of Padovan quaternion. Asian-European Journal of Mathematics 12(06), Art. No. 2040017, 8 pages.
  14. Hamilton, W. R. (1848). 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), 58–60.
  15. Horadam, A. F. (1971). Pell identities. The Fibonacci Quarterly, 9(3), 245–252.
  16. Horadam, A. F., & Mahon, J. M. (1985). Pell and Pell-Lucas polynomials. The Fibonacci Quarterly, 23(1), 7–20.
  17. Jafari, M., & Yaylı, Y. (2015). Generalized quaternions and rotation in 3-space E3αβ. TWMS Journal of Pure and Applied Mathematics, 6(2), 224–232.
  18. Jafari, M., & Yaylı, Y. (2015). Generalized quaternions and their algebraic properties. Communications Faculty of Sciences University of Ankara Series A1, 64(1), 15–27.
  19. Kalman D. (1982). Generalized Fibonacci numbers by matrix methods. The Fibonacci Quarterly, 20(1), 73–76.
  20. Khompungson, K., Rodjanadid, B., & Sompong, S. (2019). Some matrices in terms of Perrin and Padovan sequences. Thai Journal of Mathematics, 17(3), 767–774.
  21. Koshy, T. (2001). Fibonacci and Lucas Numbers with Applications. John Wiley and Sons, Inc., New York.
  22. Koshy, T. (2014). Pell and Pell–Lucas Numbers with Applications. Springer, New York.
  23. Koshy, T. (2019). Fibonacci and Lucas Numbers with Applications. John Wiley and Sons, Inc., Volume 2, New York.
  24. Mamagani, A. B., & Jafari M. (2013). On properties of generalized quaternion algebra. Journal of Novel Applied Sciences, 2(12), 683–689.
  25. Padovan R. (1994). Dom Hans van der Laan: Modern Primitive. Architectura & Natura Press, Amsterdam.
  26. Padovan, R. (2002). Dom Hans van der Laan and the Plastic Number. Nexus IV: Architecture and Mathematics, eds. Kim Williams and Jose Francisco Rodrigues. Kim Williams Books, Fucecchio (Florence), 181–193.
  27. Pottmann, H., & Wallner, J. (2001). Computational Line Geometry. Springer, New York.
  28. Seenukul, P., Netmanee S., Panyakhun, T., Auiseekaen, R., & Muangchan, S.-A. (2015). Matrices which have similar properties to Padovan Q-matrix and its generalized relations. SNRU Journal of Science and Technology, 7(2), 90–94.
  29. Shannon, A.G., Anderson, P. G., & Horadam, A. F. (2006). Properties of Cordonnier, Perrin and Van der Laan numbers. International Journal of Mathematical Education in Science and Technology, 37(7), 825–831.
  30. Shannon, A. G., & Horadam, A. F. (1972). Some properties of third-order recurrence relations. The Fibonacci Quarterly, 10(2), 135–146.
  31. Shannon, A.G., Horadam A. F., & Anderson, P. G. (2006). The auxiliary equation associated with the plastic number. Notes on Number Theory and Discrete Mathematics, 12(1), 1–12.
  32. Shannon, A. G., & Wong, C. K. (2008). Some properties of generalized third order Pell numbers. Notes on Number Theory and Discrete Mathematics, 14(4), 16–24.
  33. Sloane, N. J. A. (1964). The online encyclopedia of integer sequences, Available online at:
  34. Sokhuma, K. (2013). Matrices formula for Padovan and Perrin sequences. Applied Mathematical Sciences, 7(142), 7093–7096.
  35. Sokhuma, K. (2013). Padovan Q-matrix and the generalized relations. Applied
    Mathematical Sciences, 7(56), 2777–2780.
  36. Sompong, S., Wora-Ngon, N., Piranan, A., & Wongkaentow, N. (2017). 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, 6 pages.
  37. Soykan, Y. (2019). Simson identity of generalized m-step Fibonacci numbers. International Journal of Advances in Applied Mathematics and Mechanics, 7(2), 45–56.
  38. Soykan, Y. (2020). Generalized Pell–Padovan numbers. Asian Journal of Advanced Research and Reports, 11(2), 8–28.
  39. Soykan, Y. (2020). Summing formulas for generalized tribonacci numbers. Universal Journal of Mathematics and Applications, 3(1), 1–11.
  40. Stewart, I. (1996). Tales of a neglected number. Scientific American, 274, 102–103.
  41. Stewart, I. (2004). Math Hysteria: Fun and Games with Mathematics. Oxford University Press, New York.
  42. Taşcı, D. (2018). Padovan and Pell–Padovan quaternions. Journal of Science and Arts, 42(1), 125–132.
  43. Unger, T., & Markin, N. (2011). Quadratic forms and space-time block codes from
    generalized quaternion and biquaternion algebras. IEEE Transactions on Information Theory, 57(9), 6148–6156.
  44. Waddill, M.E., & Sacks, L. (1967). Another generalized Fibonacci sequence. The Fibonacci Quarterly, 5(3), 209–222.
  45. Yılmaz, N., & Taşkara, N. (2013). Matrix sequences in terms of Padovan and Perrin numbers. Journal of Applied Mathematics, 2013, Article ID: 941673, 7 pages.
  46. Yılmaz, N., & Taşkara, N. (2014). On the negatively subscripted Padovan and Perrin matrix sequences. Communications in Mathematics and Applications, 5(2), 59–72.

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Cite this paper

İşbilir, Z., & Gürses, N. (2021). Pell–Padovan generalized quaternions. Notes on Number Theory and Discrete Mathematics, 27(1), 171-187, doi: 10.7546/nntdm.2021.27.1.171-187.

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