On second-order linear recurrent functions with period k and proofs to two conjectures of Sroysang

Abstract

Let w be a real-valued function on R and k be a positive integer. If for every real number x, w(x + 2k) = rw(x + k) + sw(x) for some nonnegative real numbers r and s, then we call such function a second-order linear recurrent function with period k. Similarly, we call a function w : R → R satisfying w(x + 2k) = −rw(x + k) + sw(x) an odd secondorder linear recurrent function with period k. In this work, we present some elementary properties of these type of functions and develop the concept using the notion of f-even and f-odd functions discussed in [9]. We also investigate the products and quotients of these functions and provide in this work a proof of the conjecture of B. Sroysang which he posed in [19]. In fact, we offer here a proof of a more general case of the problem. Consequently, we present findings that confirm recent results in the theory of Fibonacci functions [9] and contribute new results in the development of this topic.

Keywords

• Second-order linear recurrent functions with period k,Horadam numbers,generalized Fibonacci sequences,Pell functions with period k,Jacobsthal functions with period k,Sroysang’s conjecture

References

1. Amdeberhan, T. Chen, X. Moll, V. C. and Sagan, B. E. Generalized Fibonacci polynomials and Fibonomial coefficients. Retrieved from arXiv:1306.6511v2 [math.CO] 26 Jul 2013.
2. Bacani, J. B., Rabago, J. F. T. On two nonlinear difference equations, submitted.
3. Chandra, P. and Weisstein, E. W. Fibonacci Number. From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/FibonacciNumber.html
4. Cook, C. K. Hillman, R. A. and Shannon, A. G. Some aspects of Fibonacci polynomial congruences, Ann. Math. Inform. 41, 211–217, 2013.
5. Dunlap, R. A. The golden ratio and Fibonacci numbers (World Scientific, 1998).
6. Hakami, A. An application of Fibonacci sequence on continued fractions, Int. Math. Forum 10 (2), 69–74, 2015.
7. Han, J.S. Kim, H. S. and Neggers, J. The Fibonacci norm of a positive integer n-observations and conjectures, Int. J. Number Theory 6 (2), 371–385, 2010.
8. Han, J. S. Kim, H. S. and Neggers, J. Fibonacci sequences in groupoids, Adv. Differ. Equ. 2012, Article 19, 7 pages, 2012.

9. Han, J. S. Kim, H. S. and Neggers, J. On Fibonacci functions with Fibonacci numbers, Adv. Differ. Equ. 2012, Article 126, 7 pages, 2012.
10. Horadam, A. F. Basic properties of certain generalized sequence of numbers, Fib. Quart. 3, 161–176, 1965.
11. Koshy, T. Fibonacci and Lucas numbers with applications (Pure and Applied Mathematics, Wiley-Interscience, New York, 2001).
12. Larcombe, P. J. Bagdasar, O. D. and Fennessey, E. J. Horadam sequences: a survey Bulletin of the I.C.A. 67, 49–72, 2013.
13. Larcombe, P. J. Horadam Sequences: a survey update and extension, submitted.
14. Ma, Y. and Zhang, T. On Generalized Fibonacci Polynomials and Bernoulli Numbers, J. Integer Seq. 8, Article 05.5.3, 2005.
15. Rabago, J. F. T. On solving the second-order linear recurrence sequence, Int. J. Math. Sci. Comp. 2 (1), 1–2, 2012.
16. Rabago, J. F. T. On second-order linear recurrent homogeneous differential equations with period k, Hacet. J. Math. Stat. 43 (6), 923–933, 2014.
17. Rabago, J. F. T. On k-Fibonacci numbers with applications to continued fractions, Journal of Physics: Conference Series 693 012005, 2016.
18. Rabago, J. F. T. On the closed-form solution of a nonlinear difference equation and another proof to Sroysang’s conjecture, in preparation.
19. Sroysang, B. On Fibonacci functions with period k, Discrete Dyn. Nat. Soc. 2013, Article ID 418123, 4 pages, 2013.
20. ollu, D.T. Yazlik, Y. and Taskara, N. On the solutions of two special types of Riccati difference equation via Fibonacci numbers, Adv. Differ. Equ. 2013, Article 174, 2013.
21. Vajda, S.A. Fibonacci and Lucas numbers, and the golden section (Ellis-Horwood, 1989).