#### 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.

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.

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.