Chinese Journal of MathematicsVolume 2013 (2013), Article ID 107145, 7 pageshttp://dx.doi.org/10.1155/2013/107145

Research Article

Incomplete -Fibonacci and -Lucas Numbers

José L. Ramírez

Instituto de Matemáticas y sus Aplicaciones, Universidad Sergio Arboleda, Colombia

Received 10 July 2013; Accepted 11 September 2013

Academic Editors: B. Li and W. van der Hoek

Copyright © 2013 José L. Ramírez. 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.

We define the incomplete k-Fibonacci and k-Lucas numbers; we study the recurrence relations and some properties of these numbers.

1. Introduction

Fibonacci numbers and their generalizations have many interesting properties and applications to almost every field of science and art (e.g., see [1]). Fibonacci numbers are defined by the recurrence relation

There exist a lot of properties about Fibonacci numbers. In particular, there is a beautiful combinatorial identity to Fibonacci numbers [1] From (2), Filipponi [2] introduced the incomplete Fibonacci numbers and the incomplete Lucas numbers . They are defined by

Further in [3], generating functions of the incomplete Fibonacci and Lucas numbers are determined. In [4], Djordjević gave the incomplete generalized Fibonacci and Lucas numbers. In [5], Djordjević and Srivastava defined incomplete generalized Jacobsthal and Jacobsthal-Lucas numbers. In [6], the authors define the incomplete Fibonacci and Lucas -numbers. Also the authors define the incomplete bivariate Fibonacci and Lucas -polynomials in [7].

On the other hand, many kinds of generalizations of Fibonacci numbers have been presented in the literature. In particular, a generalization is the -Fibonacci Numbers.

For any positive real number , the -Fibonacci sequence, say , is defined recurrently by

In [8], -Fibonacci numbers were found by studying the recursive application of two geometrical transformations used in the four-triangle longest-edge (4TLE) partition. These numbers have been studied in several papers; see [8–14].

For any positive real number , the -Lucas sequence, say , is defined recurrently by

If , we have the classical Lucas numbers. Moreover, ; see [15].

In [12], the explicit formula to -Fibonacci numbers is and the explicit formula of -Lucas numbers is

From (6) and (7), we introduce the incomplete -Fibonacci and -Lucas numbers and we obtain new recurrent relations, new identities, and their generating functions.

2. The Incomplete -Fibonacci Numbers

Definition 1. The incomplete -Fibonacci numbers are defined by

In Table 1, some values of incomplete -Fibonacci numbers are provided.

Table 1: The numbers , for .

0
1
2
3
4

1
1

2
k

3
k 2

4
k 3

5
k 4

6
k 5

7
k 6

8
k 7

9
k 8

10
k 9

We note that For , we get incomplete Fibonacci numbers [2].

Some special cases of (8) are

2.1. Some Recurrence Properties of the Numbers

Proposition 2. The recurrence relation of the incomplete -Fibonacci numbers is The relation (11) can be transformed into the nonhomogeneous recurrence relation

Proof. Use Definition 1 to rewrite the right-hand side of (11) as

Proposition 3. One has

Proof (by induction on ). Sum (14) clearly holds for and (see (11)). Now suppose that the result is true for all ; we prove it for :

Proposition 4. For ,

Proof (by induction on ). Sum (16) clearly holds for (see (11)). Now suppose that the result is true for all . We prove it for :

Note that if , in (4), is a real variable, then and they correspond to the Fibonacci polynomials defined by

Lemma 5. One has

See Proposition 13 of [12].

Lemma 6. One has

Proof. From (6) we have that By deriving into the previous equation (respect to ), it is obtained From Lemma 5, From where, after some algebra (20) is obtained.

Proposition 7. One has

Proof. From Lemma 6, (24) is obtained.

3. The Incomplete -Lucas Numbers

Definition 8. The incomplete -Lucas numbers are defined by

In Table 2, some numbers of incomplete -Lucas numbers are provided.

Table 2: The numbers , for .

0
1
2
3
4

1
k

2

3

4

5

6

7

8

9

We note that

Some special cases of (26) are

3.1. Some Recurrence Properties of the Numbers

Proposition 9. One has

Proof. By (8), rewrite the right-hand side of (29) as

Proposition 10. The recurrence relation of the incomplete -Lucas numbers is The relation (31) can be transformed into the nonhomogeneous recurrence relation

Proof. Using (29) and (11), we write

Proposition 11. One has

Proof. By (29), whence, from (31),

Proposition 12. One has

Proof. Using (29) and (14), we write

Proposition 13. For ,

The proof can be done by using (31) and induction on .

Lemma 14. One has

The proof is similar to Lemma 6.

Proposition 15. One has

Proof. An argument analogous to that of the proof of Proposition 7 yields From Lemma 14, (41) is obtained.

4. Generating Functions of the Incomplete -Fibonacci and -Lucas Number

In this section, we give the generating functions of incomplete -Fibonacci and -Lucas numbers.

Lemma 16 (see [3, page 592]). Let be a complex sequence satisfying the following nonhomogeneous recurrence relation: where and are complex numbers and is a given complex sequence. Then, the generating function of the sequence is where denotes the generating function of .

Theorem 17. The generating function of the incomplete -Fibonacci numbers is given by

Proof. Let be a fixed positive integer. From (8) and (12), for , , and , and Now let Also let The generating function of the sequence is (see [16, page 355]). Thus, from Lemma 16, we get the generating function of sequence .

Theorem 18. The generating function of the incomplete -Lucas numbers is given by

Proof. The proof of this theorem is similar to the proof of Theorem 17. Let be a fixed positive integer. From (26) and (32), for , , and , and Now let Also let The generating function of the sequence is (see [16, page 355]). Thus, from Lemma 16, we get the generating function of sequence .

5. Conclusion

In this paper, we introduce incomplete -Fibonacci and -Lucas numbers, and we obtain new identities. In [17], the authors introduced the -Fibonacci polynomials. That generalizes Catalan’s Fibonacci polynomials and the -Fibonacci numbers. Let be a polynomial with real coefficients. The -Fibonacci polynomials are defined by the recurrence relation

It would be interesting to study a definition of incomplete -Fibonacci polynomials and research their properties.

Acknowledgments

The author would like to thank the anonymous referees for their helpful comments. The author was partially supported by Universidad Sergio Arboleda under Grant no. USA-II-2011-0059.

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