Power series expansions of arbitrary order functional difference equations

Anthony G. Shannon, Ömür Deveci and Özgür Erdağ
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 4, Pages 851–858
DOI: 10.7546/nntdm.2025.31.4.851-858
Full paper (PDF, 387 Kb)

Details

Authors and affiliations

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

Ömür Deveci
Department of Mathematics, Faculty of Sciences and Letters, Kafkas University
36100 Kars, Türkiye

Özgür Erdağ
Department of Mathematics, Faculty of Sciences and Letters, Kafkas University
36100 Kars, Türkiye

Abstract

This paper looks at some real and complex generalizations of power series associated with some arbitrary order functional difference equations considered as generalizations and extensions of Fibonacci and Lucas numbers. It does this by drawing on, and interconnecting, some classic number theoretic results of Carlitz, Fasenmyer and Horadam.

Keywords

  • Auxiliary equation
  • Cauchy’s formula
  • Functional equations
  • Power series
  • Recursive sequence
  • Taylor series

2020 Mathematics Subject Classification

  • 11B39
  • 11C08
  • 32A05
  • 39-02

References

  1. Andrews, G. E., & Askey, R. (1977). Enumeration of partitions: The role of Eulerian series and q-orthogonal polynomials. In: Aigner, M. (Ed.). Higher Combinatorics. Boston, MA: Reidel, pp. 3–26.
  2. Bateman, H. (1936). Two systems of polynomials for the solution of Laplace’s integral equation. Duke Mathematical Journal, 2(3), 569–577.
  3. Carlitz, L. (1954). Congruences for the solution of certain difference equations of the second order. Duke Mathematical Journal, 21(4), 669–679.
  4. Carlitz, L. (1962). Generating functions for powers of certain sequences of numbers. Duke Mathematical Journal, 29(4), 521–537.
  5. Carlitz, L. (1970). Some generalized Fibonacci numbers. The Fibonacci Quarterly, 8(3), 249–254.
  6. Duffin, R. J. (1956). Basic properties of discrete analytic functions. Duke Mathematical Journal, 23(2), 335–363.
  7. Elmore, M. (1967). Fibonacci functions. The Fibonacci Quarterly, 5(4), 371–382.
  8. Fasenmyer, M. C. (1947). Some generalized hypergeometric polynomials. Bulletin of the American Mathematical Society, 53(8), 806–812.
  9. Fasenmyer, M. C. (1949.) A note on pure recurrence relations. The American Mathematical Monthly, 56(1), 14–17.
  10. Harman, C. J. (1974). A note on a discrete analytic function. Bulletin of the Australian Mathematical Society, 10(1), 123–134.
  11. Horadam, A. F. (1963). Complex Fibonacci numbers and Fibonacci quaternions. The American Mathematical Monthly, 70(3), 289–291.
  12. Hyers, D. H., Isac, G., & Rassias, T. (1998). Stability of Functional Equations in Several Variables. Birkhäuser, Basel, pp. 45–47.
  13. Jackson, F. H. (1951). Basic integration. Quarterly Journal of Mathematics. Oxford (Series 2), 2(1), 1–6.
  14. Jarden, D. (1966). Recurring Sequences: A Collection of Papers. Riveon Lematematika, Jerusalem. p. 107.
  15. Lucas, E. (1891). Théorie des Nombres. Paris: Gauthier Villars.
  16. Macmahon, P. A. (1915). Combinatory Analysis. Volume I. Cambridge University Press, Ch.1.
  17. Parizi, N. N., & Gordji, M. E. (2014). Hyers–Ulam stability of Fibonacci functional equation in modular functional spaces. Journal of Mathematics and Computer Science, 10(1), 1–6.
  18. Patel, B. K., & Ray, P. K. (2018). On the convergence of second order recurrence series. Notes on Number Theory and Discrete Mathematics, 24(4), 120–127.
  19. Rainville, E. D. (1960). Special Functions. MacMillan, New York.
  20. Shannon, A. G. (1976). Some number theoretic properties of arbitrary order recursive sequences. Bulletin of the Australian Mathematical Society, 14(1), 149–151.
  21. Shannon, A. G., Cook, C. K., & Hillman, R. A. (2017). Arbitrary order recursive sequences and associated continued fractions. Journal of Advances in Mathematics, 13(2), 7147–7154.
  22. Shannon, A. G., Loh, R. P., Melham, R. S., & Horadam, A. F. (1996). A search for solutions of a functional equation. In: Bergum, G. E., Philippou, A. N., & Horadam, A. F. (Eds.), Applications of Fibonacci Numbers, Volume 6. Dordrecht: Kluwer, pp. 431–441.
  23. Tu, S. T. (1981). Monodiffric function theory and formal power series. Journal of Mathematical Analysis and Applications, 84(2), 595–613.
  24. Whitney, R. E. (1970). On a class of difference equations. The Fibonacci Quarterly, 8(5), 470–475.
  25. Wilf, H. S. (1982). What is an answer? The American Mathematical Monthly, 89(5), 289–292.

Manuscript history

  • Received: 13 October 2025
  • Accepted: 1 November 2025
  • Online First: 19 November 2025

Copyright information

Ⓒ 2025 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).

Related papers

Cite this paper

Shannon, A. G., Deveci, Ö., & Erdağ, Ö. (2025). Power series expansions of arbitrary order functional difference equations. Notes on Number Theory and Discrete Mathematics, 31(4), 851-858, DOI: 10.7546/nntdm.2025.31.4.851-858.

Comments are closed.