SavedAdd your journal

Journals / Journal of Engineering Technology and Applied Sciences / 2026 / Cilt: 11

The Matrix Form of the k-Pell Hyperbolic Functions

k-Pell Hiperbolik Fonksiyonların Matris Formu

Efruz Özlem Mersin(AKSARAY ÜNİVERSİTESİ)0000-0001-6260-9063
Pages
—
Publication Date
2026-05-13
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

The findings presented contribute to the development of special matrix functions and their potential applications.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In the present paper, we introduce the matrix form of the k-Pell hyperbolic sine and cosine functions, along with their symmetrical forms.
📊
Key Findings & Data:Matrix functions play a significant role in various scientific fields, particularly in mathematics and engineering, as they frequently arise in the solutions of differential equations.
💡
Conclusion & Impact:The findings presented contribute to the development of special matrix functions and their potential applications.
🔬Design:Araştırma Makalesi
📍Scope:Comprehensive Sample
⚙️Variables:k-Pell sayı dizisi · Hiperbolik Foksiyonlar · Üstel Matris Fonksiyonu

Abstract

In the present paper, we introduce the matrix form of the k-Pell hyperbolic sine and cosine functions, along with their symmetrical forms. We examine their recurrence and hyperbolic properties, including Pythagorean, de Moivre, Catalan, Cassini, and d’Ocagne identities, as well as various sum and difference identities. Additionally, we define the quasi-sine k-Pell matrix functions and the matrix form of the three-dimensional k-Pell spiral, and investigate some of their fundamental properties. Matrix functions play a significant role in various scientific fields, particularly in mathematics and engineering, as they frequently arise in the solutions of differential equations. The findings presented contribute to the development of special matrix functions and their potential applications.

Özet

Mevcut makalede, k-Pell hiperbolik sinüs ve kosinüs fonksiyonlarının matris formunu, simetrik formlarıyla birlikte tanıtıyoruz. Bazı indirgeme bağıntılarını ve hiperbolik özelliklerini inceliyoruz. Matris fonksiyonları, belirli diferansiyel denklemlerin çözümlerinde göründükleri için çeşitli bilimlerde, özellikle matematik ve mühendislikte önemli bir rol oynar. Önemleri göz önüne alındığında, bu makalenin bulguları özel matris fonksiyonlarının ve potansiyel uygulamalarının ilerlemesine katkıda bulunur.

🔖

Cite This Article

Mersin, E. Ö. (2026). The Matrix Form of the k-Pell Hyperbolic Functions. Journal of Engineering Technology and Applied Sciences, Cilt: 11. https://doi.org/10.30931/jetas.1641669

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (49)

  1. Horadam, A.F., “Basic properties of a certain generalized sequence of numbers”, The Fibonacci Quarterly 3(3) (1965): 161-176.
  2. Koshy, T., “Fibonacci and Lucas numbers with applications”, Pure and Applied Mathematics, A Wiley-Interscience Series of Texts, Monographs, and Tracts, New York: Wiley, 2001.
  3. Falcón, S., Plaza, Á., “On the Fibonacci k-numbers”, Chaos, Solitons & Fractals 32(5) (2007): 1615-1624.
  4. Falcón, S., “On the k-Lucas numbers”, International Journal of Contemporary Mathematical Sciences 6(21) (2011): 1039-1050.
  5. Horadam, A.F., “Applications of modified Pell numbers to representations”, Ulam Quarterly 3(1) (1994): 34-53.
  6. Catarino, P., “On some identities and generating functions for k-Pell numbers”, International Journal of Mathematical Analysis 7(38) (2013): 1877-1884.
  7. Stakhov, A.P, Tkachenko, I.S., “Hyperbolic Fibonacci trigonometry”, Reports of the National Academy of Sciences of Ukraine 208(7) (1993): 9-14.
  8. Stakhov A, Rozin, B., “On a new class of hyperbolic functions”, Chaos, Solitons & Fractals 23(2) (2005): 379-389.
  9. Falcón, S., Plaza, Á., “The k-Fibonacci hyperbolic functions”, Chaos, Solitons & Fractals 38(2) (2008): 409-420.
  10. Taşçı, D., Azman, H., “The k-Lucas hyperbolic functions”, Communications in Mathematics and Applications 5(1) (2014): 11-21.
  11. Stakhov, A., Rozin, B., “The golden shofar”, Chaos, Solitons & Fractals 26(3) (2005): 677-684.
  12. Koçer, E.G., “Hyperbolic modified Pell functions”, Ars Combinatoria-Waterloo Then Winnipeg 85 (2007): 405-413.
  13. Koçer, E.G., Tuğlu, N., Stakhov, A., “Hyperbolic functions with second order recurrence sequences”, Ars Combinatoria 88 (2008): 65-81.
  14. Bahşi, M., Solak, S., “Hyperbolic Horadam functions”, Gazi University Journal of Science” 32(3) (2019): 956-965.
  15. Kuloğlu, B., Özkan, E., “Hyperbolic functions obtained from k-Pell sequences”, Afrika Matematika 35(1) (2024): 22.
  16. Aydın, F.T., “Hyperbolic Fibonacci sequence”, Universal Journal of Mathematics and Applications 2(2) (2019): 59-64.
  17. Kuloğlu, B., Özkan, E., “Hyperbolic functions obtained from k- Jacobsthal sequences”, Asian-European Journal of Mathematics 15(9) (2022): 2250178.
  18. Stakhov, A.P., “The generalized principle of the golden section and its applications in mathematics, science, and engineering”, Chaos, Solitons & Fractals 26(2) (2005): 263-289.
  19. Stakhov, A.P., “Hyperbolic Fibonacci and Lucas functions: A new mathematics for the living nature”, ITI, Vinnitsa, 2003.
  20. Stakhov, A., Rozin, B., “The “golden” hyperbolic models of Universe”, Chaos, Solitons & Fractals 34(2) (2007): 159-171.
  21. Bahşi, M., “Wilker-type inequalities for hyperbolic Fibonacci functions”, Journal of Inequalities and Applications 2016(1) (2016: 146.
  22. Taş, S., “The hyperbolic-type k-Fibonacci sequences and their applications”, Thermal Science 26 (Spec. issue 2) (2022): 551-558.
  23. Daşdemir, A., “On recursive hyperbolic Fibonacci quaternions”, Communications in Advanced Mathematical Sciences 4(4) (2021): 198-207.
  24. Higham, N.J., “Function of matrices: theory and computation”, Society for Industrial and Applied Mathematics, 2018.
  25. Bahşi, M., Solak, S., “On the hyperbolic Fibonacci matrix functions”, TWMS Journal of Applied and Engineering Mathematics 8(2) (2018): 454-465.
  26. Bahşi, M., Mersin, E.Ö., “On the hyperbolic Horadam matrix functions”, Hacettepe Journal of Mathematics and Statistics 51(6) (2022): 1550-1562.
  27. Horadam, A.F., “Basic properties of a certain generalized sequence of numbers”, The Fibonacci Quarterly 3(3) (1965): 161-176.
  28. Falcón, S., Plaza, Á., “On the Fibonacci k-numbers”, Chaos, Solitons & Fractals 32(5) (2007): 1615-1624.
  29. Falcón, S., “On the k-Lucas numbers”, International Journal of Contemporary Mathematical Sciences 6(21) (2011): 1039-1050.
  30. Horadam, A.F., “Applications of modified Pell numbers to representations”, Ulam Quarterly 3(1) (1994): 34-53.
  31. Catarino, P., “On some identities and generating functions for k-Pell numbers”, International Journal of Mathematical Analysis 7(38) (2013): 1877-1884.
  32. Stakhov, A.P, Tkachenko, I.S., “Hyperbolic Fibonacci trigonometry”, Reports of the National Academy of Sciences of Ukraine 208(7) (1993): 9-14.
  33. Stakhov A, Rozin, B., “On a new class of hyperbolic functions”, Chaos, Solitons & Fractals 23(2) (2005): 379-389.
  34. Falcón, S., Plaza, Á., “The k-Fibonacci hyperbolic functions”, Chaos, Solitons & Fractals 38(2) (2008): 409-420.
  35. Taşçı, D., Azman, H., “The k-Lucas hyperbolic functions”, Communications in Mathematics and Applications 5(1) (2014): 11-21.
  36. Stakhov, A., Rozin, B., “The golden shofar”, Chaos, Solitons & Fractals 26(3) (2005): 677-684.
  37. Koçer, E.G., “Hyperbolic modified Pell functions”, Ars Combinatoria-Waterloo Then Winnipeg 85 (2007): 405-413.
  38. Koçer, E.G., Tuğlu, N., Stakhov, A., “Hyperbolic functions with second order recurrence sequences”, Ars Combinatoria 88 (2008): 65-81.
  39. Bahşi, M., Solak, S., “Hyperbolic Horadam functions”, Gazi University Journal of Science” 32(3) (2019): 956-965.
  40. Kuloğlu, B., Özkan, E., “Hyperbolic functions obtained from k-Pell sequences”, Afrika Matematika 35(1) (2024): 22.
  41. Aydın, F.T., “Hyperbolic Fibonacci sequence”, Universal Journal of Mathematics and Applications 2(2) (2019): 59-64.
  42. Kuloğlu, B., Özkan, E., “Hyperbolic functions obtained from k- Jacobsthal sequences”, Asian-European Journal of Mathematics 15(9) (2022): 2250178.
  43. Stakhov, A.P., “The generalized principle of the golden section and its applications in mathematics, science, and engineering”, Chaos, Solitons & Fractals 26(2) (2005): 263-289.
  44. Stakhov, A., Rozin, B., “The “golden” hyperbolic models of Universe”, Chaos, Solitons & Fractals 34(2) (2007): 159-171.
  45. Bahşi, M., “Wilker-type inequalities for hyperbolic Fibonacci functions”, Journal of Inequalities and Applications 2016(1) (2016: 146.
  46. Taş, S., “The hyperbolic-type k-Fibonacci sequences and their applications”, Thermal Science 26 (Spec. issue 2) (2022): 551-558.
  47. Daşdemir, A., “On recursive hyperbolic Fibonacci quaternions”, Communications in Advanced Mathematical Sciences 4(4) (2021): 198-207.
  48. Bahşi, M., Solak, S., “On the hyperbolic Fibonacci matrix functions”, TWMS Journal of Applied and Engineering Mathematics 8(2) (2018): 454-465.
  49. Bahşi, M., Mersin, E.Ö., “On the hyperbolic Horadam matrix functions”, Hacettepe Journal of Mathematics and Statistics 51(6) (2022): 1550-1562.