Towards a new generalized Simson’s identity

A. G. Shannon, H. M. Srivastava and József Sàndor
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 3, Pages 479–490
DOI: 10.7546/nntdm.2024.30.3.479-490
Full paper (PDF, 870 Kb)

Details

Authors and affiliations

A. G. Shannon
¹ Warrane College, University of New South Wales, Sydney NSW 2033, Australia
² Australian Institute of Technology and Commerce, Sydney NSW 2000, Australia

H. M. Srivastava
³ Department of Mathematics and Statistics, University of Victoria
Victoria, British Columbia V8W 3R4, Canada
⁴ Department of Medical Research, China Medical University Hospital,
China Medical University, Taichung 40402, Taiwan

József Sàndor
⁵ Department of Mathematics, Babeş-Bolyai University, Cluj-Napoca 400347, Romania

Abstract

This paper is an attempt to develop an elegant and simple generalization of what is usually called Simson’s Identity, with variations named after Cassini, Catalan and Gelin-Cesàro. It can shed a new light on Simson’s identity, and possibly how to extend it to some reciprocals of these identities and how to generalize it to arbitrary order with some conjectures.

Keywords

• Fibonacci and Lucas numbers
• Recurrence relations
• Riemann Zeta Function
• Simson’s Identity
• Kronecker delta

2020 Mathematics Subject Classification

• 11B39
• 11B0F

References

1. Atanassov, K. T., Atanassova, V., Shannon, A. G., & Turner, J. C. (2002). New Visual Perspectives on Fibonacci Numbers. Singapore/New Jersey/London/Hong Kong: World Scientific Publishing Company.
2. Bell, E. T. (1924). Notes on recurring series of the third order. Tôhoku Mathematical Journal, 24, 168–184.
3. Bergum, G. E. (1984). Addenda to Geometry of a generalized Simson’s formula. The Fibonacci Quarterly, 22(1), 22–28.
4. Carlitz, L. (1960). Some arithmetic sums connected with the greatest integer function. Mathematica Scandinavica. 8(1), 59–64.
5. Dickson, L. E. (1952). History of the Theory of Numbers. Volume 1. New York: Chelsea.
6. Frontczak, R., Srivastava, H. M., & Tomovski, Ž. (2021). Some families of Apéry-like Fibonacci and Lucas series. Mathematics, 6(14), Article ID 1621.
7. Gootherts, G. W. (1968). A linear algebra constructed from Fibonacci sequences, Part I: Fundamentals and polynomial interpretations. The Fibonacci Quarterly, 6(1), 35–43.
8. Horadam, A. F. (1965). Basic properties of a certain generalized sequence of numbers. The Fibonacci Quarterly, 3(3), 161–176.
9. Horadam, A. F. (1982). Geometry of a generalized Simson’s formula. The Fibonacci Quarterly, 20(2), 164–168.
10. Horadam, A. F., & Treweek, A. P. (1986). Simson’s formula and an equation of degree 24. The Fibonacci Quarterly, 24(4), 344–346.
11. Knuth, D. (1997). The Art of Computer Programming, Vol. 1: Fundamental Algorithms. (3rd Ed.). Upper Saddle River, NJ: Addison-Wesley, p. 81.
12. Lucas, E. (1878). Théorie des fonctions numériques simplement périodiques. American Journal of Mathematics, 1, 184–240, 289–321.
13. Macmahon, P. A. (1915). Combinatory Analysis, Volume I. Cambridge/London/New York: Cambridge University Press.
14. Mangon, J. (1885). Étude sur la théorie du tir. Revie Militaire Belge. 10(3), 5–51; Continuation: (1885), ibid, 10(4), 163–186; (1886), ibid, 11(4), 5–46.
15. Melham, R. S. (2003). A Fibonacci Identity in the spirit of Simson and Gelin-Cesàro. The Fibonacci Quarterly, 41(2), 142–143,
16. Melham, R. S., & Shannon, A. G. (1995). A generalization of the Catalan identity and some consequences. The Fibonacci Quarterly, 33(1), 82–84.
17. Shannon, A. G. (1974). Some properties of a fundamental recursive sequence of arbitrary order. The Fibonacci Quarterly, 12(4), 327–335.
18. Shannon, A. G. (2013). The Sequences of Horadam, Williams and Philippou as generalized Lucas sequences. Advanced Studies in Contemporary Mathematics, 23(3), 551–558.
19. Sloane, N. J. A. (2024). The Online Encyclopedia of Integer Sequences. Available online at: https://oeis.org.
20. Tuenter, H. J. H. (2022). Fibonacci summation identities arising from Catalan’s identity. The Fibonacci Quarterly, 60(4), 312–319.
21. Vajda, S. (1989). Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications. Mineola, NY: Dover, Ch.III.
22. Van der Poorten, A. J. (1973). A note on recurrence sequences. Journal and Proceedings of the Royal Society of New South Wales, 106, 115–117.
23. Williams, H. C. (1972). On a generalization of the Lucas functions. Acta Arithmetica, 20(1), 33–51.
24. Williams, H. C. (1972). Fibonacci numbers obtained from Pascal’s triangle. The Fibonacci Quarterly, 10(4), 405–412.

Manuscript history

• Received: 16 June 2024
• Revised: 11 September 2024
• Accepted: 11 September 2024
• Online First: 18 September 2024

Copyright information

Ⓒ 2024 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).