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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 840345, 11 pages
http://dx.doi.org/10.1155/2012/840345
Review Article

Incomplete Bivariate Fibonacci and Lucas -Polynomials

1Department of Mathematics, Faculty of Science, Gazi University, Teknikokullar, 06500 Ankara, Turkey
2Department of Mathematics, Faculty of Education, Başkent University, Baglica, 06810 Ankara, Turkey

Received 26 November 2011; Accepted 8 February 2012

Academic Editor: Gerald Teschl

Copyright © 2012 Dursun Tasci et al. 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 bivariate Fibonacci and Lucas polynomials. In the case , , we obtain the incomplete Fibonacci and Lucas numbers. If , , we have the incomplete Pell and Pell-Lucas numbers. On choosing , , we get the incomplete generalized Jacobsthal number and besides for the incomplete generalized Jacobsthal-Lucas numbers. In the case , , , we have the incomplete Fibonacci and Lucas numbers. If , , , , we obtain the Fibonacci and Lucas numbers. Also generating function and properties of the incomplete bivariate Fibonacci and Lucas polynomials are given.

1. Introduction

Djordjević introduced incomplete generalized Fibonacci and Lucas numbers using explicit formulas of generalized Fibonacci and Lucas numbers in [1]. In [2] incomplete Fibonacci and Lucas numbers are given as follows: where . Note that for the case incomplete Fibonacci numbers are reduced to Fibonacci numbers and for the case incomplete Lucas numbers are reduced to Lucas numbers in [2]. Also the authors considered the generating functions of the incomplete Fibonacci and Lucas numbers in [3]. In [4] Djordjević and Srivastava defined incomplete generalized Jacobsthal and Jacobsthal-Lucas numbers.

The generalized Fibonacci and Lucas -numbers were studied in [5, 6]. Incomplete Fibonacci and Lucas -numbers are defined by for in [7]. In [8] the authors introduced incomplete Pell and Pell-Lucas -numbers.

The generalized bivariate Fibonacci -polynomials and generalized bivariate Lucas -polynomials are defined the recursion for with and with in [5]. When , . In [5], the authors obtained some relations for these polynomials sequences. In addition, in [5], the explicit formula of bivariate Fibonacci -polynomials is and the explicit formula of bivariate Lucas -polynomials is In this paper, we defined incomplete bivariate Fibonacci and Lucas -polynomials. We generalize incomplete Fibonacci and Lucas numbers, incomplete generalized Fibonacci numbers, incomplete generalized Jacobsthal numbers, incomplete Fibonacci and Lucas -numbers, incomplete Pell and Pell-Lucas -numbers.

2. Incomplete Bivariate Fibonacci and Lucas -Polynomials

Definition 2.1. For , , incomplete bivariate Fibonacci -polynomials are defined as

For , , , we get incomplete Fibonacci -numbers [7].

If , , , we obtained incomplete Pell -numbers [8].

On choosing , , , we have incomplete generalized Jacobsthal numbers [4].

If , , , , we get incomplete Fibonacci numbers [2].

For , we obtained Fibonacci numbers [9].

Definition 2.2. For , , incomplete bivariate Lucas -polynomials are defined as

If , , , we obtained incomplete Lucas -numbers [7].

For , , , we have incomplete Pell-Lucas -numbers [8].

On choosing , , , , we get incomplete generalized Jacobsthal-Lucas numbers [4].

If , , , , we obtained incomplete Lucas numbers [2].

For , , , , we have Lucas numbers [9].

Proposition 2.3. The incomplete bivariate Fibonacci -polynomials satisfy the following recurrence relation:

Proof. Using (2.1), we obtain

Taking in (2.3), we could obtain a formula for incomplete Fibonacci -numbers (see [7, Proposition  3]). Taking in (2.3), we could obtain a formula for incomplete Fibonacci numbers (see [2, Proposition  1]).

Proposition 2.4. The nonhomogeneous recurrence relation of incomplete bivariate Fibonacci -polynomials is

Proof. It is easy to obtain from (2.1) and (2.3).

Proposition 2.5. For , one has

Proof. Equation (2.6) clearly holds for . Suppose that the equation holds for . We show that the equation holds for . We have

Proposition 2.6. For , 

Proof. Equation (2.8) can be easily proved by using (2.3) and induction on .

We have the following proposition in which the relationship between the incomplete bivariate Fibonacci and Lucas -polynomials is preserved as found in [5] before.

Proposition 2.7. One has

Proof. By (2.1), rewrite the right-hand side of (2.9) as

Proposition 2.8. The incomplete bivariate Lucas -polynomials satisfy the following recurrence relation:

Proof. We write by using (2.3) and (2.9)

Proposition 2.9. The nonhomogeneous recurrence relation of incomplete bivariate Lucas -polynomials is

Proof. The proof can be done by using (2.2) and (2.11).

Proposition 2.10. For , one has

Proof. Proof is similar to the proof of Proposition 2.5.

Proposition 2.11. For , one has

Proof. Proof is obtained immediately by using (2.11) and induction .

Proposition 2.12. One has

Proof. We can write from (2.2) Equation (2.17) is calculated using the formula and [5]

Then we have the following conclusion.

Conclusion. When in (2.16), we obtain which is Proposition  11 in [2].

3. Generating Functions of the Incomplete Bivariate Fibonacci and Lucas -Polynomials

Lemma 3.1 (see [3]). Let be a complex sequence satisfying the following nonhomogeneous recurrence relation: where is a given complex sequence. Then the generating function of the sequence is where denotes the generating function of .

Theorem 3.2. The generating function of the incomplete bivariate Fibonacci -polynomials is

Proof. From (2.1) and (2.5), ,  and for Now let and Also We obtained that is the generating function of the sequence . From Lemma  3.1, we get that the generating function of sequence is  Therefore,

Theorem 3.3. The generating function of the incomplete bivariate Lucas -polynomials is

Proof. From (2.9) and (3.3),
For the general case in Theorems  3.2 and 3.3, we find the generating functions of some special numbers by the special cases . For example, in (3.3) we obtain the generating function of incomplete Fibonacci -numbers.

References

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