We define the incomplete bivariate Fibonacci and Lucas p-polynomials. In the case x = 1, y = 1, we obtain the incomplete Fibonacci and Lucas p-numbers. If x = 2, y = 1, we have the incomplete Pell and Pell-Lucas p-numbers. On choosing x = 1, y = 2, we get the incomplete generalized Jacobsthal number and besides for p = 1 the incomplete generalized Jacobsthal-Lucas numbers. In the case x = 1, y = 1, p = 1, we have the incomplete Fibonacci and Lucas numbers. If x = 1, y = 1, p = 1, k = ⌊(n − 1)/(p + 1)⌋, we obtain the Fibonacci and Lucas numbers. Also generating function and properties of the incomplete bivariate Fibonacci and Lucas p-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:

(1.1)

where n = 1, 2, 3, …. Note that for the case k = ⌊(n − 1)/2⌋ incomplete Fibonacci numbers are reduced to Fibonacci numbers and for the case k = ⌊n/2⌋ 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 p-numbers were studied in [5, 6]. Incomplete Fibonacci and Lucas p-numbers are defined by

(1.2)

for n ≥ 1 in [7]. In [8] the authors introduced incomplete Pell and Pell-Lucas p-numbers.

The generalized bivariate Fibonacci p-polynomials F_p,n(x, y) and generalized bivariate Lucas p-polynomials L_p,n(x, y) are defined the recursion for p ≥ 1

(1.3)

with

(1.4)

and

(1.5)

with

(1.6)

in [5]. When x = y = 1, F_p,n(1,1) = F_p(n). In [5], the authors obtained some relations for these polynomials sequences. In addition, in [5], the explicit formula of bivariate Fibonacci p-polynomials is

(1.7)

and the explicit formula of bivariate Lucas p-polynomials is

(1.8)

In this paper, we defined incomplete bivariate Fibonacci and Lucas p-polynomials. We generalize incomplete Fibonacci and Lucas numbers, incomplete generalized Fibonacci numbers, incomplete generalized Jacobsthal numbers, incomplete Fibonacci and Lucas p-numbers, incomplete Pell and Pell-Lucas p-numbers.

2. Incomplete Bivariate Fibonacci and Lucas p-Polynomials

Definition 2.1. For p ≥ 1, n ≥ 1, incomplete bivariate Fibonacci p-polynomials are defined as

(2.1)

For x = 1, y = 1, , we get incomplete Fibonacci p-numbers [7].

If x = 2, y = 1, , we obtained incomplete Pell p-numbers [8].

On choosing x = 1, y = 2, , we have incomplete generalized Jacobsthal numbers [4].

If x = 1, y = 1, p = 1, , we get incomplete Fibonacci numbers [2].

For , we obtained Fibonacci numbers [9].

Definition 2.2. For p ≥ 1, n ≥ 1, incomplete bivariate Lucas p-polynomials are defined as

(2.2)

If x = 1, y = 1, , we obtained incomplete Lucas p-numbers [7].

For x = 2, y = 1, , we have incomplete Pell-Lucas p-numbers [8].

On choosing x = 1, y = 2, p = 1, , we get incomplete generalized Jacobsthal-Lucas numbers [4].

If x = 1, y = 1, p = 1, , we obtained incomplete Lucas numbers [2].

For x = 1, y = 1, p = 1, k = ⌊n/(p + 1)⌋, , we have Lucas numbers [9].

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

(2.3)

Proof. Using (2.1), we obtain

(2.4)

Taking x = y = 1 in (2.3), we could obtain a formula for incomplete Fibonacci p-numbers (see [7, Proposition 3]). Taking x = y = p = 1 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 p-polynomials is

(2.5)

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

Proposition 2.5. For 0 ≤ k ≤ (n − h − p − 1)/(p + 1), one has

(2.6)

Proof. Equation (2.6) clearly holds for h = 0. Suppose that the equation holds for h > 0. We show that the equation holds for (h + 1). We have

(2.7)

Proposition 2.6. For n ≥ k(p + 1) + p + 2,

(2.8)

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

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

Proposition 2.7. One has

(2.9)

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

(2.10)

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

(2.11)

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

(2.12)

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

(2.13)

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

Proposition 2.10. For 0 ≤ k ≤ (n − p − h)/(p + 1), one has

(2.14)

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

Proposition 2.11. For n ≥ (k + 1)(p + 1), one has

(2.15)

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

Proposition 2.12. One has

(2.16)

Proof. We can write from (2.2)

(2.17)

Equation (2.17) is calculated using the formula L_p,n(x, y) and ∂L_p,n(x, y)/∂x = nF_p,n(x, y) [5]

(2.18)

Then we have the following conclusion.

Conclusion. When x = y = p = 1 in (2.16), we obtain

(2.19)

which is Proposition 11 in [2].

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

Lemma 3.1 (see [3].)Let be a complex sequence satisfying the following nonhomogeneous recurrence relation:

(3.1)

where {r_n} is a given complex sequence. Then the generating function

of the sequence {s_n} is

(3.2)

where G(t) denotes the generating function of {r_n}.

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

(3.3)

Proof. From (2.1) and (2.5), ,

(3.4)

and for n ≥ k(p + 1) + p + 2

(3.5)

Now let

(3.6)

and

(3.7)

Also

(3.8)

We obtained that G(t) = y^k+1t^p+1/(1−xt)^k+1 is the generating function of the sequence {r_n}. From Lemma 3.1, we get that the generating function

of sequence {s_n} is

(3.9)

Therefore,

(3.10)

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

(3.11)

Proof. From (2.9) and (3.3),

(3.12)

For the general case in Theorems 3.2 and 3.3, we find the generating functions of some special numbers by the special cases x, y, p. For example, x = y = 1 in (3.3) we obtain the generating function of incomplete Fibonacci p-numbers.

References

1 Djordjević G. B., Generating functions of the incomplete generalized Fibonacci and generalized Lucas numbers, The Fibonacci Quarterly. (2004) 42, no. 2, 106–113, 2071775, ZBL1053.11009.
Web of Science® Google Scholar
2 Filipponi P., Incomplete Fibonacci and Lucas numbers, Rendiconti del Circolo Matematico di Palermo. Serie II. (1996) 45, no. 1, 37–56, https://doi.org/10.1007/BF02845088, 1389739, ZBL0855.11007.
10.1007/BF02845088
Google Scholar
3 Pintér Á. and Srivastava H. M., Generating functions of the incomplete Fibonacci and Lucas numbers, Rendiconti del Circolo Matematico di Palermo. Serie II. (1999) 48, no. 3, 591–596, https://doi.org/10.1007/BF02844348, 1731459, ZBL0934.11008.
10.1007/BF02844348
Google Scholar
4 Djordjević G. B. and Srivastava H. M., Incomplete generalized Jacobsthal and Jacobsthal-Lucas numbers, Mathematical and Computer Modelling. (2005) 42, no. 9-10, 1049–1056, https://doi.org/10.1016/j.mcm.2004.10.026, 2181299, ZBL1178.11015.
10.1016/j.mcm.2004.10.026
Google Scholar
5 Tuglu N., Kocer E. G., and Stakhov A., Bivariate Fibonacci like p-polynomials, Applied Mathematics and Computation. (2011) 217, no. 24, 10239–10246, https://doi.org/10.1016/j.amc.2011.05.022, 2806410.
10.1016/j.amc.2011.05.022
Web of Science® Google Scholar
6 Kocer E. G. and Tuglu N., The Binet formulas for the Pell and Pell-Lucas p-numbers, Ars Combinatoria. (2007) 85, 3–17, 2359276.
Web of Science® Google Scholar
7 Tasci D. and Firengiz M. C., Incomplete Fibonacci and Lucas p-numbers, Mathematical and Computer Modelling. (2010) 52, no. 9-10, 1763–1770, https://doi.org/10.1016/j.mcm.2010.07.003, 2719591.
10.1016/j.mcm.2010.07.003
Google Scholar
8 Tasci D., Cetin Firengiz M., and Djordjević G. B., Incomplete Pell and Pell-Lucas p-numbers, Submit.
Google Scholar
9 Koshy T., Fibonacci and Lucas Numbers with Applications, 2001, Wiley, New York, NY, USA, Pure and Applied Mathematics, 1855020.
10.1002/9781118033067
Google Scholar

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Incomplete Bivariate Fibonacci and Lucas p-Polynomials

Abstract

1. Introduction

2. Incomplete Bivariate Fibonacci and Lucas p-Polynomials

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

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Incomplete Bivariate Fibonacci and Lucas p-Polynomials

Abstract

1. Introduction

2. Incomplete Bivariate Fibonacci and Lucas p-Polynomials

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

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