On Third-Order Linear Recurrent Functions

Abstract

A function is said to be a Tribonacci function with period if , for all . In this paper, we present some properties on the Tribonacci functions with period . We show that if is a Tribonacci function with period , then , where is the root of the equation such that .

1. Introduction

The most popular numbers studied in many different forms for centuries are Fibonacci numbers. Fibonacci sequence is famous for their amazing properties (see [1–5]). Many research articles talk about these numbers. The third-order linear recurrence of these sequences are what we call Tribonacci sequences by the recurrence relation with , , , and as an integer. Han, Kim, and Neggers studied Fibonacci numbers [6–8] and introduced the concept of Fibonacci functions with Fibonacci numbers in [8] which were later extended by B. Sroysang [9] to Fibonacci functions with period . In the same order as in [8], Parizi and Gordji [10] studied Tribonacci functions. They gave some properties of Tribonacci functions: a function is said to be a Tribonacci function if , for all . They also showed that if is a Tribonacci function, then which is a root of .

In this paper, for any positive integer , a function is said to be a Tribonacci function with period if , for all . In Sections 2 and 3, we present some properties of these functions. In Section 4, we develop the notions of these functions using the concept of even and odd functions discussed in [8, 10]. We also show that if is a Tribonacci function with period , then .

2. Tribonacci Functions with Period

In this section, we present some properties of Tribonacci functions with period .

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

Example 2. Let be a positive root of the equation . Then, . Define a map by , where . Then, is a Tribonacci function with period .

Proposition 3. Let be a Tribonacci function with period and define , for all , where . Then, is also a Tribonacci function with period .

Proof. Let . Then,

Theorem 4. Let be a Tribonacci function with period and let , , and be the sequences of Tribonacci numbers with and and . Then, for any and an integer.

Proof. . The assertion holds for
So, fix and assume that the assertion is valid for every . Then,

Corollary 5. Let , , and be the sequences of Tribonacci numbers with and , , and , . Let be the root of the equation such that . Then, .

Proof. Let . We have seen in Example 2 that is a Tribonacci function with period . Applying Theorem 4, we have , for all . We obtain .

Remark 6. Consider the Tribonacci sequenses , , and as in Theorem 4. We have the relations and .

3. Odd Tribonacci Functions with Period

Here we discuss odd Tribonacci functions with period defined as follows.

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

Example 8. Let be an odd Tribonacci function with period , . It is clear that , for all . We have . Then, such that is root of the equation , . Thus, is an odd Tribonacci function with period on .

Proposition 9. Let be an odd Tribonacci function with period and define , for all , where . Then, is also an odd Tribonacci function with period .

Proof. Let . Then,

Theorem 10. Let be an odd Tribonacci function with period and let , , and be the sequences of Tribonacci numbers with and and . Then, for any .

Proof. . The assertion holds for
So, fix and assume that the assertion is valid for every . Then,

4. Even and Odd Functions with Period

In this section, we will talk about the notion of Tribonacci functions using even and odd functions. Here we get results obtained in [10] with third-order linear recurrence. We give the limit of the quotient of a Tribonacci function with period , extending the results of [11] in third-order linear recurrence.

Definition 11. Let be a positive integer and a function such that the preimage of by has the empty interior. The function is said to be an even (resp., odd) function with period if (resp., ), for all .

Example 12. If , then implies that if . Due to the fact that is dense in and is continuous, it follows that . Let and . Then, = = = Hence, is an even function with period .

Theorem 13. Let and be an even function with period and let be a continuous function. Then, is a (resp., an odd) Tribonacci function with period if and only if is a (resp., an odd) Tribonacci function with period .

Proof. Let be a Tribonacci function with period . For any , we haveHence, is a Tribonacci function with period .
Now assume that is a Tribonacci function with period . For , we haveFor all , we get . Hence, is a Tribonacci function with period .

We give the proof for the case where is a Tribonacci function with period . The case where is an odd Tribonacci function with period is similar and left to the reader.

Now we give the limit of the quotient of a Tribonacci function with period .

Theorem 14. Let and be a Tribonacci function with period . Then, the limit of the quotient is , the root of the equation such that .

Proof. Given , there exists and such that . Let us set and . By Theorem 4, we have and hence . Hence,By Remark 6, we haveNow, the well-known result that yieldsHence, we obtain

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that he has no conflicts of interest.

Acknowledgments

The author would like to thank the referees for their helpful comments and remarks.