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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 418123, 4 pages
http://dx.doi.org/10.1155/2013/418123
Research Article

On Fibonacci Functions with Period

Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathumthani 12121, Thailand

Received 6 February 2013; Accepted 4 April 2013

Academic Editor: Gabriele Bonanno

Copyright © 2013 Banyat Sroysang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A function is said to be a Fibonacci function if for all . In 2012, some properties on the Fibonacci functions were presented. In this paper, for any positive integer , a function is said to be a Fibonacci function with period if for all ; we present some properties on the Fibonacci functions with period .

1. Introduction

Presently, there are many research articles about Fibonacci numbers (see [1]). Fibonacci numbers are also involved in the golden ratio (see [2]). In 2008, Kim and Neggers [3] studied Fibonacci means. In 2009, Jung [4] studied Hyers-Ulam stability of Fibonacci functional equation. In 2010, Han et al. [5] studied a Fibonacci norm of positive integers. In 2012, Han et al. [6] studied Fibonacci sequences in groupoids. Moreover, they [7] gave some properties on Fibonacci functions; a function is said to be a Fibonacci function if , for all , using the concept of -even and -odd functions. They also showed that if is a Fibonacci function, then .

In this paper, for any positive integer , a function is said to be a Fibonacci function with period if for all ; we present some properties on the Fibonacci functions with period using the concept of -even and -odd functions with period . Moreover, we also present some properties on the odd Fibonacci functions with period .

2. Fibonacci Functions with Period

Definition 1. Let be a positive integer. A function is said to be a Fibonacci function with period if for all .

Example 2. Let be a Fibonacci function with period , where . It follows that for all , so . Then . Thus, for all .

Proposition 3. Let be a Fibonacci function with period . Assume that is differentiable. Then is also a Fibonacci function with period .

Proof. Let . Since , it follows that .

Proposition 4. Let be a Fibonacci function with period , and define for all , where . Then is also a Fibonacci function with period .

Proof. Let . Then .

Example 5. Let and . Define by for all . Then is a Fibonacci function with period .

Theorem 6. Let be a Fibonacci function with period , and let be a sequence of Fibonacci numbers with , , and for all . Then, for any and , .

Proof. Let . We note that and . Now, we assume that and , where . Then This proof is completed.

3. Odd Fibonacci Functions with Period

Definition 7. Let be a positive integer. A function is said to be an odd Fibonacci function with period if for all .

Example 8. Let be an odd Fibonacci function with period , where . It follows that for all , so . Then . Thus, for all .

Proposition 9. Let be an odd Fibonacci function with period . Assume that is differentiable. Then is also an odd Fibonacci function with period .

Proof. Let . Since , it follows that .

Proposition 10. Let be an odd Fibonacci function with period , and define for all , where . Then is also an odd Fibonacci function with period .

Proof. Let . Then .

Example 11. Let and . Define by for all . Then is an odd Fibonacci function with period .

Theorem 12. Let be an odd Fibonacci function with period , and let be a sequence of Fibonucci numbers with , , and for all . Then, for any and , .

Proof. Let . We note that and . Now, we assume that and , where . Then This proof is completed.

4. -Even Functions with Period

Definition 13. Let be a positive integer and let be such that if , where is continuous, then . The function is said to be an -even function with period if for all .

Example 14. Define for all . Let be a continuous function such that . For any , we have , so . Since is dense in and is continuous, it follows that . Let and . Then . Hence, is an -even function with period .

Theorem 15. Let and be an -even function with period and let be a continuous function. Then is a Fibonacci function with period if and only if is a Fibonacci function with period .

Proof. First, we assume that is a Fibonacci function with period . For any , we have
Hence, is a Fibonacci function with period .
Next, we assume that is a Fibonacci function with period . Let . Then
By the assumption of , we obtain that . Hence, is a Fibonacci function with period .

Example 16. Let . Define and for all . For all , we have . We recall that is an -even function with period , and is a Fibonacci function with period . Hence, is a Fibonacci function with period .

Theorem 17. Let and be an -even function with period and let be a continuous function. Then is an odd Fibonacci function with period if and only if is an odd Fibonacci function with period .

Proof. First, we assume that is an odd Fibonacci function with period . For any , we have
Hence, is an odd Fibonacci function with period .
Next, we assume that is an odd Fibonacci function with period . Let . Then
By the assumption of , we obtain that . Hence, is an odd Fibonacci function with period .

Example 18. Let . Define and for all . For all , we have . We recall that is an -even function with period and is an odd Fibonacci function with period . Hence, is an odd Fibonacci function with period .

5. -Odd Functions with Period

Definition 19. Let be a positive integer and let be such that if where is continuous, then . The function is said to be an -odd function with period if for all .

Example 20. Define for all . Let be a continuous function such that . For any , we have , so . Since is dense in and is continuous, it follows that . Let be a positive odd integer and . Then . Hence, is an -even function with period .

Theorem 21. Let and be an -odd function with period and let be a continuous function. Then is a Fibonacci function with period if and only if is an odd Fibonacci function with period .

Proof. First, we assume that is a Fibonacci function with period . For any , we have
Hence, is an odd Fibonacci function with period .
Next, we assume that is an odd Fibonacci function with period . Let . Then
By the assumption of , we obtain that . Hence, is a Fibonacci function with period .

Example 22. Let be a positive odd integer. Define and for all . We have for all . We recall that is an -odd function with period and is a Fibonacci function with period . Hence, is an odd Fibonacci function with period .

Theorem 23. Let and be an -odd function with period and let be a continuous function. Then is an odd Fibonacci function with period if and only if is a Fibonacci function with period .

Proof. First, we assume that is an odd Fibonacci function with period . For any , we have
Hence, is a Fibonacci function with period .
Next, we assume that is a Fibonacci function with period . Let . Then
By the assumption of , we obtain that . Hence, is an odd Fibonacci function with period .

Example 24. Let be a positive odd integer. Define and for all . We have for all . We recall that is an -odd function with period and is an odd Fibonacci function with period . Hence, is a Fibonacci function with period .

6. Open Problems

Conjecture 25. If is a Fibonacci function with period , then

Conjecture 26. If is an odd Fibonacci function with period , then

Acknowledgments

The author would like to thank the referees for their useful comments and suggestions.

References

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