Some aspects of interchanging difference equation orders

Anthony G. Shannon and Engin Özkan
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 3, Pages 507–516
DOI: 10.7546/nntdm.2022.28.3.507-516
Full paper (PDF, 592 Kb)

Details

Authors and affiliations

Anthony G. Shannon
Warrane College, University of New South Wales
Kensington, NSW 2033, Australia

Engin Özkan
Department of Mathematics, Erzincan Binali Yildirim University
Erzincan, Turkey

Abstract

This paper builds on Roettger and Williams’ extensions of the primordial Lucas sequence to consider some relations among difference equations of different orders. This paper utilises some of their second and third order recurrence relations to provide an excursion through basic second order sequences and related third order recurrence relations with a variety of numerical illustrations which demonstrate that mathematical notation is a tool of thought.

Keywords

• Arbitrary order recurrence relations
• Primordial sequences
• Vandermonde determinants

2020 Mathematics Subject Classification

• 11B37
• 11B39
• 11B50

References

1. Atanassov, K. T., & Shannon, A. G. (1998). Matrix-tertions and matrix-noitrets: exercises in mathematical enrichment. International Journal of Mathematical Education in Science and Technology, 29(6), 898–903.
2. Bell, E. T. (1924). Notes on recurring sequences of the third order. Tohoku Mathematical Journal, 24, 168–174.
3. Carlitz, L. (1960). Some arithmetic sums connected with the greatest integer function. Mathematica Scandinavica, 8(1), 59–64.
4. Dickson, L. E. (1952). History of the Theory of Numbers. Volume 1, Chelsea, New York.
5. Feinberg, M. (1963). Fibonacci–Tribonacci. The Fibonacci Quarterly, 1(3), 71–74.
6. Frontczak, R. (2019). Relations for generalized Fibonacci and Tribonacci sequences. Notes on Number Theory and Discrete Mathematics, 25(1), 178–192.
7. Gonzalez, M. F., Vandebrouck, F., & Vivier, L. (2022). A classic recursive sequence calculus task at the secondary-tertiary level in France. International Journal of Mathematical Education in Science and Technology, 53(5), 1092–1112.
8. Gootherts, G. W. (1968). A linear algebra constructed from Fibonacci sequences, Part I: Fundamentals and polynomial interpretations. The Fibonacci Quarterly, 6(1), 35–43.
9. Griffiths, M., & Bramham, A. (2015). The Jacobsthal Numbers. Two Results and Two Questions. The Fibonacci Quarterly, 53(2), 147–151.
10. Hoggatt, V. E. Jr. (1969). Fibonacci and Lucas Numbers. Houghton-Mifflin Boston, MA.
11. Horadam, A. F. (1965). Basic properties of a certain generalized sequence of numbers. The Fibonacci Quarterly, 13(3), 161–176.
12. Horadam, A. F. (1967). Special properties of the sequence W_n(a,b;p,q). The Fibonacci Quarterly, 5(4), 424–434.
13. Horadam, A. F. (1971). Pell Identities. The Fibonacci Quarterly, 9(3), 245–263.
14. Hunter, J. (1964). Number Theory. Oliver and Boyd, Edinburgh, pp. 22–23.
15. Iverson, K. E. (1980). Notation as a Tool of Thought. Communications of the Association for Computing Machinery, 23(8), 444–465.
16. Jarden, D. (1966). Recurring Sequences. Riveon Lematimatika, Jerusalem, p. 107.
17. Khomovsky, D. I. (2016). Efficient computation of terms of linear recurrence sequences of any order. Integers, 18, Article ID A39.
18. Larcombe, P. J., & Fennessey, E. J. (2018). A new non-linear recurrence identity class for Horadam sequence terms. Palestine Journal of Mathematics, 7(2), 406–409.
19. Lucas, É. (1878). Théorie des fonctions numériques simplement périodiques. American Journal of Mathematics, 1, 184–240, 288–321.
20. Müller, S., Williams, H. C., & Roettger, E. (2009). A cubic extension of the Lucas functions. Annales des Sciences Mathématiques du Québec, 33(2), 185–224.
21. Ocke, K. (1997). Factoring integers defined by second and third order recurrence relations. Master of Science thesis, Computer Science Department, Rochester Institute of Technology.
22. Roettger, E. L., & Williams, H. C. (2012). Public-key cryptography based on a cubic extension of the Lucas functions. Fundamenta Informaticae, 114(3-4), 325–344.
23. Roettger, E. L., & Williams, H. C. (2021). Some primality tests constructed from a cubic extension of the Lucas functions. The Fibonacci Quarterly, 59(3), 194–213.
24. Roettger, E. L., Williams, H. C., & Guy, R. K. (2015). Some primality that eluded Lucas. Designs, Codes and Cryptography, 77(2), 515–539.
25. Shannon, A. G. (1972). Iterative formulas associated with generalized third order recurrence relations. Society for Industrial and Applied Mathematics Journal on Applied Mathematics, 23(3), 364–368.
26. Shannon, A. G. (1974). Some properties of a fundamental recursive sequence of arbitrary order. The Fibonacci Quarterly, 12(4), 327–335.
27. 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.
28. Shannon, A. G., & Horadam, A.F. (1971). Generating functions for powers of third order recurrence sequences. Duke Mathematical Journal, 38(4), 791–794.
29. Vorob’ev, N. N. (1961). Fibonacci Numbers. Pergamon, Oxford.
30. Williams, H. C. (1972). On a generalization of the Lucas functions. Acta Arithmetica, 20, 33–51.
31. Williams, H. C. (1972). Fibonacci numbers obtained from Pascal’s triangle with generalizations. The Fibonacci Quarterly, 10(4), 405–412.
32. Williams, H. C. (1998). Édouard Lucas and Primality Testing. In: Canadian Mathematical Society Series of Monographs and Advanced Texts. Vol. 22. Wiley-Interscience, New York.
33. Yakibu, G., Bernard, M. L., Lucy, B. G., & Domven, L. (2022). A secured cryptographic technique using rhotrices in polygraphic cyber systems. Science Forum, 22(1), 35–42.
34. Young, D. M., & Gregory, R. T. (1972). A Survey of Numerical Mathematics. Addison-Wesley, Boston, MA.

Manuscript history

• Received: 6 June 2022
• Revised: 28 July 2022
• Accepted: 3 August 2022
• Online First: 4 August 2022