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.