Research Article  Open Access
José L. Ramírez, "Incomplete Fibonacci and Lucas Numbers", Chinese Journal of Mathematics, vol. 2013, Article ID 107145, 7 pages, 2013. https://doi.org/10.1155/2013/107145
Incomplete Fibonacci and Lucas Numbers
Abstract
We define the incomplete kFibonacci and kLucas 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 JacobsthalLucas 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 fourtriangle longestedge (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 righthand 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 righthand 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. USAII20110059.
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Copyright
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.