SavedAdd your journal

Journals / Gazi University Journal of Science / 2026 / Volume: 39 Issue: 3

Exploring Harmonic and Hyper-harmonic Narayana Numbers through Circulant Structures and Matrix Norms

Bahar Kuloǧlu(SİVAS BİLİM VE TEKNOLOJİ ÜNİVERSİTESİ)0000-0001-7624-8270
Pages
—
Publication Date
2026-09-01
🏛️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 results contribute both to the combinatorial theory of special number sequences and to the normative analysis of structured matrices.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:This study investigates the structural and analytical properties of harmonic and hyper-harmonic Narayana numbers, extending classical results on harmonic and hyper-harmonic numbers.
📊
Key Findings & Data:Furthermore, we examine the spectral and Euclidean norms of circulant and r-circulant matrices generated by these numbers, together with selected special matrix forms.
💡
Conclusion & Impact:The results contribute both to the combinatorial theory of special number sequences and to the normative analysis of structured matrices.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Narayana sequence · Harmonic Narayana number · Hyper-harmonic- Narayana number

Abstract

This study investigates the structural and analytical properties of harmonic and hyper-harmonic Narayana numbers, extending classical results on harmonic and hyper-harmonic numbers. We establish novel combinatorial identities and derive closed-form expressions for and ​, which represent finite sums involving reciprocals of Narayana numbers. Furthermore, we examine the spectral and Euclidean norms of circulant and r-circulant matrices generated by these numbers, together with selected special matrix forms. Through this framework, several inequalities associated with matrix norms are obtained, providing deeper insight into the interplay between Narayana-type sequences and matrix analysis. The results contribute both to the combinatorial theory of special number sequences and to the normative analysis of structured matrices.

🔖

Cite This Article

Kuloǧlu, B. (2026). Exploring Harmonic and Hyper-harmonic Narayana Numbers through Circulant Structures and Matrix Norms. Gazi University Journal of Science, Volume: 39 Issue: 3. https://doi.org/10.35378/gujs.1788997

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (39)

  1. Allouche, J.-P., and Johnson, T., "Narayana’s cows and delayed morphisms", Troisièmes Journées d’Informatique Musicale (JIM ’96): 2-7, (1996).
  2. Özkan, E., and Kuloğlu, B., "On the new Narayana polynomials, the Gauss Narayana numbers and their polynomials", Asian-European Journal of Mathematics, 14(6): 2150100, (2021). DOI: https://doi.org/10.1142/S179355712150100X.
  3. Ramírez, J.L., and Sirvent, V.F., "A note on the k-Narayana sequence", Annales Mathematicae et Informaticae, 45: 91-105, (2015).
  4. Kapoor, S., and Kumar, P., "On harmonic complex Narayana-Lucas sequences and harmonic hybrid Narayana-Lucas sequences", Communications in Combinatorics and Optimization, Articles in Press, (2025). DOI: https://doi.org/10.22049/cco.2025.30102.2309.
  5. Özkan, E., Kuloğlu, B., and Peters, J.F., "K-Narayana sequence self-similarity: flip graph views of k-Narayana self-similarity", Chaos, Solitons & Fractals, 153: 111473, (2021). DOI: https://doi.org/10.1016/j.chaos.2021.111473.
  6. Conway, J.H., and Guy, R.K., "The Book of Numbers", Copernicus, New York, (1996). DOI: https://doi.org/10.1007/978-1-4612-4072-3.
  7. Graham, R.L., Knuth, D.E., and Patashnik, O., "Concrete Mathematics: A Foundation for Computer Science", 2nd ed., Addison-Wesley, Reading, (1994). DOI: https://doi.org/10.5555/562056.
  8. Ohtsuka, H., and Nakamura, S., "On the sum of reciprocal Fibonacci numbers", The Fibonacci Quarterly, 46/47(2): 153-159, (2008/2009). DOI: https://doi.org/10.1080/00150517.2008.12428174.
  9. Holliday, S.H., and Komatsu, T., "On the sum of reciprocal generalized Fibonacci numbers", Integers, 11(4): 441-455, (2011). DOI: https://doi.org/10.1515/integ.2011.031.
  10. Tuğlu, N., Kızılateş, C., and Kesim, S., "On the harmonic and hyperharmonic Fibonacci numbers", Advances in Difference Equations, 2015: 297, (2015). DOI: https://doi.org/10.1186/s13662-015-0635-z.
  11. Çetin, M., Kızılateş, C., Yeşil Baran, F., and Tuğlu, N., "Some identities of harmonic and hyperharmonic Fibonacci numbers", Gazi University Journal of Science, 34(2): 493-504, (2021). DOI: https://doi.org/10.35378/gujs.705885.
  12. Dişkaya, O., and Menken, H., "Harmonic and hyperharmonic Padovan numbers", Research & Reviews in Science and Mathematics, Gece Kitaplığı, Ankara, 167-178, (2022). ISBN: 978-625-430-441-5.
  13. Kim, D.S., and Kim, T., "Identities involving harmonic and hyperharmonic numbers", Advances in Difference Equations, 2013: 235, (2013). DOI: https://doi.org/10.1186/1687-1847-2013-235.
  14. Duran, O., Ömür, N., and Koparal, S., "On sums with generalized harmonic, hyperharmonic and special numbers", Miskolc Mathematical Notes, 21(2): 791-803, (2020). DOI: https://doi.org/10.18514/MMN.2020.3458.
  15. Rim, S.-H., Kim, T., and Pyo, S.-S., "Identities between harmonic, hyperharmonic and Daehee numbers", Journal of Inequalities and Applications, 2018: 168, (2018). DOI: https://doi.org/10.1186/s13660-018-1757-0.
  16. Anđelić, M., da Fonseca, C.M., Kızılateş, C., and Terzioğlu, N., "r-min and r-max matrices with harmonic higher order Gauss Fibonacci numbers entries", Journal of Applied Mathematics and Computing, 71: 7437-7461, (2025). DOI: https://doi.org/10.1007/s12190-025-02530-z.
  17. Anđelić, M., da Fonseca, C.M., Kızılateş, C., and Terzioğlu, N., "Properties of the harmonic and hyperharmonic higher order Gauss Fibonacci numbers", Mathematical Methods in the Applied Sciences, 49(1): 457-465, (2026). DOI: https://doi.org/10.1002/mma.70164.
  18. Ingleton, A.W., "The rank of circulant matrices", Journal of the London Mathematical Society, s1-31(4): 445-460, (1956). DOI: https://doi.org/10.1112/jlms/s1-31.4.445.
  19. Davis, P.J., "Circulant Matrices", John Wiley & Sons, New York, (1979). ISBN: 0-471-05771-1.
  20. Golub, G.H., and Van Loan, C.F., "Matrix Computations", 3rd ed., Johns Hopkins University Press, Baltimore, (1996). ISBN: 978-0-8018-5414-9.
  21. Kumari, M., Prasad, K., Tanti, J., and Özkan, E., "On the properties of r-circulant matrices involving Mersenne and Fermat numbers", International Journal of Nonlinear Analysis and Applications, 14(5): 121-131, (2023). DOI: https://doi.org/10.22075/ijnaa.2023.27875.3742.
  22. Kuloğlu, B., Eser, E., and Özkan, E., "The r-circulant matrices associated with k-Fermat and k-Mersenne numbers", WSEAS Transactions on Mathematics, 22: 531-543, (2023). DOI: https://doi.org/10.37394/23206.2023.22.59.
  23. Kuloğlu, B., Eser, E., and Özkan, E., "On the properties of r-circulant matrices involving generalized Fermat numbers", Sakarya University Journal of Science, 27(5): 956-964, (2023). DOI: https://doi.org/10.16984/saufenbilder.1280572.
  24. Falcon, S., "On the circulant k-Fibonacci matrices", IOSR Journal of Mathematics, 13(2): 38-42, (2017). DOI: https://doi.org/10.9790/5728-1302023842.
  25. Uygun, Ş., and Yaşamalı, S., "On the bounds for the norms of r-circulant matrices with the Jacobsthal and Jacobsthal-Lucas numbers", International Journal of Pure and Applied Mathematics, 112(1): 93-102, (2017). DOI: https://doi.org/10.12732/ijpam.v112i1.7.
  26. Bozkurt, D., and Tam, T.-Y., "Determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas numbers", Applied Mathematics and Computation, 219(2): 544-551, (2012). DOI: https://doi.org/10.1016/j.amc.2012.06.039.
  27. Bueno, A.C.F., "On the eigenvalues and the determinant of the right circulant matrices with Pell and Pell-Lucas numbers", International Journal of Mathematics and Scientific Computing, 4(1): 19-20, (2014).
  28. Solak, S., "On the norms of circulant matrices with the Fibonacci and Lucas numbers", Applied Mathematics and Computation, 160(1): 125-132, (2005). DOI: https://doi.org/10.1016/j.amc.2003.08.126.
  29. Zheng, Y., and Shon, S., "Exact inverse matrices of Fermat and Mersenne circulant matrix", Abstract and Applied Analysis, 2015: Article ID 760823, 10 pages, (2015). DOI: https://doi.org/10.1155/2015/760823.
  30. Shen, S.Q., and Cen, J.M., "On the bounds for the norms of r-circulant matrices with the Fibonacci and Lucas numbers", Applied Mathematics and Computation, 216(10): 2891-2897, (2010). DOI: https://doi.org/10.1016/j.amc.2010.03.140.
  31. Tuğlu, N., and Kızılateş, C., "On the norms of some special matrices with the harmonic Fibonacci numbers", Gazi University Journal of Science, 28(3): 497-501, (2015).
  32. Tuğlu, N., and Kızılateş, C., "On the norms of circulant and r-circulant matrices with the hyperharmonic Fibonacci numbers", Journal of Inequalities and Applications, 2015: 253, (2015). DOI: https://doi.org/10.1186/s13660-015-0778-1.
  33. Horn, R.A., and Johnson, C.R., "Matrix Analysis", Cambridge University Press, Cambridge, (1985). DOI: https://doi.org/10.1017/CBO9780511810817.
  34. Kocer, E.G., Mansour, T., and Tuğlu, N., "Norms of circulant and semicirculant matrices with Horadam’s numbers", Ars Combinatoria, 85: 353-359, (2007).
  35. Zhou, J., "The identical estimates of spectral norms for circulant matrices with binomial coefficients combined with Fibonacci numbers and Lucas numbers entries", Journal of Function Spaces, 2014: Article ID 672398, 5 pages, (2014). DOI: https://doi.org/10.1155/2014/672398.
  36. Bahşı, M., and Solak, S., "On the norms of r-circulant matrices with the hyper-Fibonacci and Lucas numbers", Journal of Mathematical Inequalities, 8(4): 693-705, (2014). DOI: https://doi.org/10.7153/jmi-08-52.
  37. Jiang, Z., and Zhou, J., "A note on spectral norms of even-order r-circulant matrices", Applied Mathematics and Computation, 250: 368-371, (2015). DOI: https://doi.org/10.1016/j.amc.2014.11.020.
  38. Zhou, J., Chen, X., and Jiang, Z., "The explicit identities for spectral norms of circulant-type matrices involving binomial coefficients and harmonic numbers", Mathematical Problems in Engineering, 2014: Article ID 518913, 7 pages, (2014). DOI: https://doi.org/10.1155/2014/518913.
  39. Graham, R.L., Knuth, D.E., and Patashnik, O., "Concrete Mathematics: A Foundation for Computer Science", Addison-Wesley, Reading, (1989). ISBN: 0-201-14236-8.