Incomplete Bivariate Fibonacci and Lucas 𝑝-Polynomials

Abstract

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: 𝐹_𝑛(𝑘)=^𝑘_𝑗=0𝑗𝑛−1−𝑗,0≤𝑘≤𝑛−12,𝐿_𝑛(𝑘)=^𝑘_𝑗=0𝑛𝑗𝑛𝑛−𝑗𝑛−𝑗,0≤𝑘≤2, (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 𝐹^𝑘_𝑝(𝑛)=^𝑘_𝑗=0𝑗𝑛−𝑗𝑝−1,0≤𝑘≤𝑛−1,𝐿𝑝+1_𝑘𝑝(𝑛)=^𝑘_𝑗=0𝑛𝑗𝑛𝑛−𝑗𝑝𝑛−𝑗𝑝,0≤𝑘≤,𝑝+1 (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≥1𝐹_𝑝,𝑛𝑥,𝑦=𝑥𝐹_{𝑝,𝑛−1}𝑥,𝑦+𝑦𝐹_{𝑝,𝑛−𝑝−1}𝑥,𝑦,𝑛>𝑝, (1.3) with𝐹_𝑝,0𝑥,𝑦=0,𝐹_𝑝,𝑛𝑥,𝑦=𝑥^𝑛−1for𝑛=1,2,…𝑝, (1.4) and 𝐿_𝑝,𝑛𝑥,𝑦=𝑥𝐿_{𝑝,𝑛−1}𝑥,𝑦+𝑦𝐿_{𝑝,𝑛−𝑝−1}𝑥,𝑦,𝑛>𝑝, (1.5) with 𝐿_𝑝,0𝑥,𝑦=𝑝+1,𝐿_𝑝,𝑛𝑥,𝑦=𝑥^𝑛for𝑛=1,2,…𝑝 (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 𝐹_𝑝,𝑛𝑥,𝑦=^{⌊(𝑛−1)/(𝑝+1)⌋}_𝑗=0𝑗𝑥𝑛−𝑗𝑝−1_{𝑛−𝑗(𝑝+1)−1}𝑦^𝑗,𝑛≥0,𝑝≥1, (1.7) and the explicit formula of bivariate Lucas 𝑝-polynomials is 𝐿_𝑝,𝑛𝑥,𝑦=^{⌊𝑛/(𝑝+1)⌋}_𝑗=0𝑛𝑗𝑥𝑛−𝑗𝑝𝑛−𝑗𝑝_{𝑛−𝑗(𝑝+1)}𝑦^𝑗,𝑛≥0,𝑝≥1. (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 𝐹^𝑘_𝑝,𝑛𝑥,𝑦=^𝑘_𝑗=0𝑗𝑥𝑛−𝑗𝑝−1_{𝑛−𝑗(𝑝+1)−1}𝑦^𝑗,0≤𝑘≤𝑛−1.𝑝+1 (2.1)

For 𝑥=1, 𝑦=1, 𝐹^𝑘_𝑝,𝑛(𝑥,𝑦)=𝐹^𝑘_𝑝(𝑛), we get incomplete Fibonacci 𝑝-numbers [7].

If 𝑥=2, 𝑦=1, 𝐹^𝑘_𝑝,𝑛(𝑥,𝑦)=𝑃^𝑘_𝑝(𝑛), we obtained incomplete Pell 𝑝-numbers [8].

On choosing 𝑥=1, 𝑦=2, 𝐹^𝑘_𝑝,𝑛(𝑥,𝑦)=𝐽^𝑘_𝑛,𝑝+1, we have incomplete generalized Jacobsthal numbers [4].

If 𝑥=1, 𝑦=1, 𝑝=1, 𝐹^𝑘_𝑝,𝑛(𝑥,𝑦)=𝐹_𝑛(𝑘), we get incomplete Fibonacci numbers [2].

For 𝑥=1,𝑦=1,𝑝=1,𝑘=⌊(𝑛−1)/(𝑝+1)⌋,𝐹^𝑘_𝑝,𝑛(𝑥,𝑦)=𝐹_𝑛, we obtained Fibonacci numbers [9].

Definition 2.2. For 𝑝≥1, 𝑛≥1, incomplete bivariate Lucas 𝑝-polynomials are defined as 𝐿^𝑘_𝑝,𝑛𝑥,𝑦=^𝑘_𝑗=0𝑛𝑗𝑥𝑛−𝑗𝑝𝑛−𝑗𝑝_{𝑛−𝑗(𝑝+1)}𝑦^𝑗𝑛,0≤𝑘≤.𝑝+1 (2.2)

If 𝑥=1, 𝑦=1, 𝐿^𝑘_𝑝,𝑛(𝑥,𝑦)=𝐿^𝑘_𝑝(𝑛), we obtained incomplete Lucas 𝑝-numbers [7].

For 𝑥=2, 𝑦=1, 𝐿^𝑘_𝑝,𝑛(𝑥,𝑦)=𝑄^𝑘_𝑝(𝑛), we have incomplete Pell-Lucas 𝑝-numbers [8].

On choosing 𝑥=1, 𝑦=2, 𝑝=1, 𝐿^𝑘_𝑝,𝑛(𝑥,𝑦)=𝑗^𝑘_𝑛,𝑝+1, we get incomplete generalized Jacobsthal-Lucas numbers [4].

If 𝑥=1, 𝑦=1, 𝑝=1, 𝐿^𝑘_𝑝,𝑛(𝑥,𝑦)=𝐿_𝑛(𝑘), we obtained incomplete Lucas numbers [2].

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

Proposition 2.3. The incomplete bivariate Fibonacci 𝑝-polynomials satisfy the following recurrence relation: 𝐹^𝑘+1_𝑝,𝑛𝑥,𝑦=𝑥𝐹^𝑘+1_{𝑝,𝑛−1}𝑥,𝑦+𝑦𝐹^𝑘_{𝑝,𝑛−𝑝−1}𝑥,𝑦,0≤𝑘≤𝑛−𝑝−3.𝑝+1 (2.3)

Proof. Using (2.1), we obtain 𝑥𝐹^𝑘+1_{𝑝,𝑛−1}𝑥,𝑦+𝑦𝐹^𝑘_{𝑝,𝑛−𝑝−1}𝑥,𝑦=𝑥^𝑘+1_𝑗=0𝑗𝑥𝑛−𝑗𝑝−2_{𝑛−𝑗(𝑝+1)−2}𝑦^𝑗+𝑦^𝑘_𝑗=0𝑗𝑥𝑛−𝑝−𝑝𝑗−2_{𝑛−𝑝−𝑗(𝑝+1)−2}𝑦^𝑗=^𝑘+1_𝑗=0𝑗𝑥𝑛−𝑗𝑝−2_{𝑛−𝑗(𝑝+1)−1}𝑦^𝑗+^𝑘_𝑗=0𝑗𝑥𝑛−𝑝−𝑝𝑗−2_{𝑛−𝑝−𝑗(𝑝+1)−2}𝑦^𝑗+1=^𝑘+1_𝑗=0𝑗𝑥𝑛−𝑗𝑝−2_{𝑛−𝑗(𝑝+1)−1}𝑦^𝑗+^𝑘+1_𝑗=1𝑥𝑛−𝑝𝑗−2𝑗−1^{𝑛−𝑗(𝑝+1)−1}𝑦^𝑗=^𝑘+1_𝑗=0𝑗+𝑥𝑛−𝑗𝑝−2𝑛−𝑝𝑗−2𝑗−1^{𝑛−𝑗(𝑝+1)−1}𝑦^𝑗−𝑥𝑛−2−1^𝑛−1=^𝑘+1_𝑗=0𝑗𝑥𝑛−𝑗𝑝−1_{𝑛−𝑗(𝑝+1)−1}𝑦^𝑗−0=𝐹^𝑘+1_𝑝,𝑛𝑥,𝑦. (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 𝐹^𝑘_𝑝,𝑛𝑥,𝑦=𝑥𝐹^𝑘_{𝑝,𝑛−1}𝑥,𝑦+𝑦𝐹^𝑘_{𝑝,𝑛−𝑝−1}−()𝑘𝑥,𝑦𝑥𝑛−𝑝𝑘+1−2_{𝑛−𝑝(𝑘+1)−𝑘−2}𝑦^𝑘+1. (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 ^ℎ_𝑗=0ℎ𝑗𝑦^ℎ−𝑗𝑥^𝑗𝐹^𝑘+𝑗_{𝑝,𝑛+𝑝𝑗−1}𝑥,𝑦=𝐹^𝑘+ℎ_{𝑝,𝑛+𝑝+1ℎ−𝑝}𝑥,𝑦. (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 ^ℎ+1𝑗=0𝑗𝑦ℎ+1_ℎ+1−𝑗𝑥^𝑗𝐹^𝑘+𝑗_{𝑝,𝑛+𝑝𝑗−1}=𝑥,𝑦_ℎ+1𝑗=0ℎ𝑗+ℎ𝑦𝑗−1^ℎ+1−𝑗𝑥^𝑗𝐹^𝑘+𝑗_{𝑝,𝑛+𝑝𝑗−1}=𝑥,𝑦_ℎ+1𝑗=0ℎ𝑗𝑦^ℎ+1−𝑗𝑥^𝑗𝐹^𝑘+𝑗_{𝑝,𝑛+𝑝𝑗−1}𝑥,𝑦+^ℎ+1_𝑗=0ℎ𝑦𝑗−1^ℎ+1−𝑗𝑥^𝑗𝐹^𝑘+𝑗_{𝑝,𝑛+𝑝𝑗−1}𝑥,𝑦=𝑦𝐹^𝑘+ℎ_{𝑝,𝑛+𝑝+1ℎ−𝑝}+ℎ𝑥,𝑦𝑥ℎ+1^ℎ+1𝐹^𝑘+ℎ+1_{𝑝,𝑛+𝑝ℎ}+𝑥,𝑦_ℎ𝑗=−1ℎ𝑗𝑦^ℎ−𝑗𝑥^𝑗+1𝐹^𝑘+𝑗+1_{𝑝,𝑛+𝑝𝑗}𝑥,𝑦=𝑦𝐹^𝑘+ℎ_{𝑝,𝑛+𝑝+1ℎ−𝑝}𝑥,𝑦+𝑥^ℎ_𝑗=0ℎ𝑗𝑦^ℎ−𝑗𝑥^𝑗𝐹^𝑘+𝑗+1_{𝑝,𝑛+𝑝𝑗}+ℎ𝑦𝑥,𝑦−1^ℎ+1𝐹^𝑘_{𝑝,𝑛−𝑝}𝑥,𝑦=𝑦𝐹^𝑘+ℎ_{𝑝,𝑛+𝑝+1ℎ−𝑝}𝑥,𝑦+𝑥𝐹_ℎ^𝑘+ℎ+1_{𝑝,𝑛+𝑝+1}𝑥,𝑦=𝐹^𝑘+ℎ+1_{𝑝,𝑛+𝑝+1ℎ+1}𝑥,𝑦. (2.7)

Proposition 2.6. For 𝑛≥𝑘(𝑝+1)+𝑝+2,  ^ℎ−1_𝑗=0𝑦𝑥^𝑗𝐹^𝑘_{𝑝,𝑛−𝑝+𝑗}=1𝑥,𝑦𝑥^ℎ−1𝐹^𝑘+1_{𝑝,𝑛+ℎ}𝑥,𝑦−𝑥𝐹^𝑘+1_𝑝,𝑛𝑥,𝑦. (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 𝐿^𝑘_𝑝,𝑛𝑥,𝑦=𝐹^𝑘_𝑝,𝑛+1𝑥,𝑦+𝑝𝑦𝐹^𝑘−1_{𝑝,𝑛−𝑝}𝑛𝑥,𝑦,0≤𝑘≤.𝑝+1 (2.9)

Proof. By (2.1), rewrite the right-hand side of (2.9) as 𝐹^𝑘_𝑝,𝑛+1𝑥,𝑦+𝑝𝑦𝐹^𝑘−1_{𝑝,𝑛−𝑝}𝑥,𝑦=^𝑘_𝑗=0𝑗𝑥𝑛−𝑗𝑝_{𝑛−𝑗(𝑝+1)}𝑦^𝑗+𝑝𝑦^𝑘−1_𝑗=0𝑗𝑥𝑛−𝑝−𝑗𝑝−1_{𝑛−𝑝−𝑗(𝑝+1)−1}𝑦^𝑗=^𝑘_𝑗=0𝑗𝑥𝑛−𝑗𝑝_{𝑛−𝑗(𝑝+1)}𝑦^𝑗+𝑝𝑦^𝑘_𝑗=1𝑥𝑛−𝑗𝑝−1𝑗−1^{𝑛−𝑗(𝑝+1)}𝑦^𝑗−1=^𝑘_𝑗=0𝑗+𝑥𝑛−𝑗𝑝𝑛−𝑗𝑝−1𝑗−1^{𝑛−𝑗(𝑝+1)}𝑦^𝑗−𝑥𝑛−1−1^𝑛=^𝑘_𝑗=0𝑛𝑗𝑥𝑛−𝑗𝑝𝑛−𝑗𝑝_{𝑛−𝑗(𝑝+1)}𝑦^𝑗=𝐿^𝑘_𝑝,𝑛𝑥,𝑦. (2.10)

Proposition 2.8. The incomplete bivariate Lucas 𝑝-polynomials satisfy the following recurrence relation: 𝐿^𝑘+1_𝑝,𝑛𝑥,𝑦=𝑥𝐿^𝑘+1_{𝑝,𝑛−1}𝑥,𝑦+𝑦𝐿^𝑘_{𝑝,𝑛−𝑝−1}𝑥,𝑦,0≤𝑘≤𝑛−𝑝−2.𝑝+1 (2.11)

Proof. We write by using (2.3) and (2.9) 𝐿^𝑘+1_𝑝,𝑛𝑥,𝑦=𝐹^𝑘+1_𝑝,𝑛+1𝑥,𝑦+𝑝𝑦𝐹^𝑘_{𝑝,𝑛−𝑝}𝑥,𝑦=𝑥𝐹^𝑘+1_𝑝,𝑛𝑥,𝑦+𝑦𝐹^𝑘_{𝑝,𝑛−𝑝}𝑥,𝑦+𝑝𝑦𝑥𝐹^𝑘_{𝑝,𝑛−𝑝−1}𝑥,𝑦+𝑦𝐹^𝑘−1_{𝑝,𝑛−2𝑝−1}𝐹𝑥,𝑦=𝑥^𝑘+1_𝑝,𝑛𝑥,𝑦+𝑝𝑦𝐹^𝑘_{𝑝,𝑛−𝑝−1}𝑥,𝑦+𝑦𝐹^𝑘_{𝑝,𝑛−𝑝}𝑥,𝑦+𝑝𝑦𝐹^𝑘−1_{𝑝,𝑛−2𝑝−1}𝑥,𝑦=𝑥𝐿^𝑘+1_{𝑝,𝑛−1}𝑥,𝑦+𝑦𝐿^𝑘_{𝑝,𝑛−𝑝−1}𝑥,𝑦. (2.12)

Proposition 2.9. The nonhomogeneous recurrence relation of incomplete bivariate Lucas 𝑝-polynomials is 𝐿^𝑘_𝑝,𝑛𝑥,𝑦=𝑥𝐿^𝑘_{𝑝,𝑛−1}𝑥,𝑦+𝑦𝐿^𝑘_{𝑝,𝑛−𝑝−1}−𝑥,𝑦𝑛−𝑝−1()()𝑘𝑥𝑛−𝑝𝑘+1−1𝑛−𝑝𝑘+1−1_{𝑛−(𝑝+1)(𝑘+1)}𝑦^𝑘+1. (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 ^ℎ_𝑗=0ℎ𝑗𝑥^𝑗𝑦^ℎ−𝑗𝐿^𝑘+𝑗_{𝑝,𝑛+𝑝𝑗−1}𝑥,𝑦=𝐿^𝑘+ℎ_{𝑝,𝑛+𝑝+1ℎ−𝑝}𝑥,𝑦. (2.14)

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

Proposition 2.11. For 𝑛≥(𝑘+1)(𝑝+1), one has ^ℎ−1_𝑗=0𝑦𝑥^𝑗𝐿^𝑘_{𝑝,𝑛−𝑝+𝑗}=1𝑥,𝑦𝑥^ℎ−1𝐿^𝑘+1_{𝑝,𝑛+ℎ}𝑥,𝑦−𝑥𝐿^𝑘+1_𝑝,𝑛𝑥,𝑦. (2.15)

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

Proposition 2.12. One has ^{⌊𝑛/(𝑝+1)⌋}_𝑘=0𝐿^𝑘_𝑝,𝑛=𝑛𝑥,𝑦𝐿𝑝+1+1_𝑝,𝑛+𝑛𝑥,𝑦𝑝+1𝑥𝐹_𝑝,𝑛𝑥,𝑦−𝐿_𝑝,𝑛𝑥,𝑦. (2.16)

Proof. We can write from (2.2) ^{⌊𝑛/(𝑝+1)⌋}_𝑘=0𝐿^𝑘_𝑝,𝑛𝑥,𝑦=𝐿⁰_𝑝,𝑛𝑥,𝑦+𝐿¹_𝑝,𝑛𝑥,𝑦+𝐿²_𝑝,𝑛𝑥,𝑦+⋯+𝐿^{⌊𝑛/(𝑝+1)⌋}_𝑝,𝑛=𝑛𝑥,𝑦𝑛𝑛0𝑥^𝑛+𝑛𝑛𝑛0𝑥^𝑛+𝑛1𝑥𝑛−𝑝𝑛−𝑝_{𝑛−(𝑝+1)}𝑦+𝑛𝑛𝑛0𝑥^𝑛+𝑛1𝑥𝑛−𝑝𝑛−𝑝_{𝑛−(𝑝+1)}𝑛𝑦+2𝑥𝑛−2𝑝𝑛−2𝑝_{𝑛−2(𝑝+1)}𝑦²+𝑛+⋯𝑛𝑛0𝑥^𝑛𝑛+⋯+𝑝×𝑛𝑛−𝑛/𝑝+1𝑛−𝑝𝑛𝑝+1𝑥𝑝+1^{𝑛−(𝑝+1)⌊𝑛/(𝑝+1)⌋}𝑦^{⌊𝑛/(𝑝+1)⌋}=𝑛𝑛𝑝+1+1𝑛𝑛0𝑥^𝑛+𝑛𝑛𝑝+1+1−11𝑥𝑛−𝑝𝑛−𝑝_{𝑛−(𝑝+1)}𝑦+𝑛𝑛𝑝+1+1−21𝑥𝑛−2𝑝𝑛−2𝑝_{𝑛−2(𝑝+1)}𝑦²+𝑛+⋯𝑝𝑛𝑛−𝑛/𝑝+1𝑛−𝑝𝑛𝑝+1𝑥𝑝+1^{𝑛−(𝑝+1)⌊𝑛/(𝑝+1)⌋}𝑦^{⌊𝑛/(𝑝+1)⌋}=^{⌊𝑛/(𝑝+1)⌋}_𝑗=0𝑛𝑛𝑝+1+1−𝑗𝑗𝑥𝑛−𝑗𝑝𝑛−𝑗𝑝_{𝑛−𝑗(𝑝+1)}𝑦^𝑗=𝑛𝑝+1+1^{⌊𝑛/(𝑝+1)⌋}_𝑗=0𝑛𝑗𝑥𝑛−𝑗𝑝𝑛−𝑗𝑝_{𝑛−𝑗(𝑝+1)}𝑦^𝑗−^{⌊𝑛/(𝑝+1)⌋}_𝑗=0𝑗𝑛𝑗𝑥𝑛−𝑗𝑝𝑛−𝑗𝑝_{𝑛−𝑗(𝑝+1)}𝑦^𝑗. (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] ^{⌊𝑛/(𝑝+1)⌋}_𝑘=0𝐿^𝑘_𝑝,𝑛=𝑛𝑥,𝑦𝑝+1+1^{⌊𝑛/(𝑝+1)⌋}_𝑗=0𝑛𝑗𝑥𝑛−𝑗𝑝𝑛−𝑗𝑝_{𝑛−𝑗(𝑝+1)}𝑦^𝑗+𝑛𝐹_𝑝,𝑛𝑥,𝑦−𝑛𝑥⁻¹𝐿_𝑝,𝑛𝑥,𝑦𝑝+1𝑥⁻¹=𝑛𝐿𝑝+1+1_𝑝,𝑛+𝑛𝑥,𝑦𝑝+1𝑥𝐹_𝑝,𝑛𝑥,𝑦−𝐿_𝑝,𝑛𝑥,𝑦. (2.18)

Then we have the following conclusion.

Conclusion. When 𝑥=𝑦=𝑝=1 in (2.16), we obtain ^⌊𝑛/2⌋_𝑘=0𝐿_𝑛(𝑘)=𝑛2+1𝐿_𝑛+𝑛2𝐹_𝑛−𝐿_𝑛 (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 {𝑠_𝑛}^∞_𝑛=0 be a complex sequence satisfying the following nonhomogeneous recurrence relation: 𝑠_𝑛=𝑥𝑠_𝑛−1+𝑦𝑠_{𝑛−𝑝−1}+𝑟_𝑛,𝑛>𝑝, (3.1) where {𝑟_𝑛} is a given complex sequence. Then the generating function 𝑆^𝑘_𝑝(𝑥,𝑦;𝑡) of the sequence {𝑠_𝑛} is 𝑆^𝑘_𝑝𝑥,𝑦;𝑡=𝑠₀−𝑟₀+^𝑝_𝑖=1𝑠_𝑖−𝑥𝑠_𝑖−1−𝑟_𝑖𝑡^𝑖+𝐺(𝑡)1−𝑥𝑡−𝑦𝑡^𝑝+1−1, (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 𝑅^𝑘_𝑝𝑥,𝑦;𝑡=𝑡^{𝑘(𝑝+1)+1}𝐹_{𝑝,𝑘(𝑝+1)+1}𝑥,𝑦+^𝑝_𝑖=1𝑡^𝑖𝐹_{𝑝,𝑘(𝑝+1)+1+𝑖}𝑥,𝑦−𝑥𝐹_{𝑝,𝑘(𝑝+1)+𝑖}+𝑦𝑥,𝑦_𝑘+1𝑡^𝑝+1(1−𝑥𝑡)^𝑘+11−𝑥𝑡−𝑦𝑡^𝑝+1−1. (3.3)

Proof. From (2.1) and (2.5), 𝐹^𝑘_𝑝,𝑛(𝑥,𝑦)=0for0≤𝑛<𝑘(𝑝+1)+1,  𝐹^𝑘_{𝑝,𝑘𝑝+1+1}𝑥,𝑦=𝐹_{𝑝,𝑘(𝑝+1)+1},𝐹𝑥,𝑦_{𝑘𝑝,𝑘𝑝+1+2}𝑥,𝑦=𝐹_{𝑝,𝑘(𝑝+1)+2},⋮𝐹𝑥,𝑦_{𝑘𝑝,𝑘𝑝+1+𝑝+1}𝑥,𝑦=𝐹_{𝑝,𝑘(𝑝+1)+𝑝+1}𝑥,𝑦, (3.4) and for 𝑛≥𝑘(𝑝+1)+𝑝+2  𝐹^𝑘_𝑝,𝑛𝑥,𝑦=𝑥𝐹^𝑘_{𝑝,𝑛−1}𝑥,𝑦+𝑦𝐹^𝑘_{𝑝,𝑛−𝑝−1}−()𝑥,𝑦𝑥𝑛−𝑝𝑘+1−2𝑛−𝑘𝑝+1−𝑝−2^{𝑛−𝑝(𝑘+1)−𝑘−2}𝑦^𝑘+1. (3.5) Now let 𝑠₀=𝐹^𝑘_{𝑝,𝑘𝑝+1+1}𝑥,𝑦,𝑠₁=𝐹^𝑘_{𝑝,𝑘𝑝+1+2}𝑥,𝑦,…,𝑠_𝑝=𝐹^𝑘_{𝑝,𝑘𝑝+1+𝑝+1}𝑥,𝑦, (3.6) and 𝑠_𝑛=𝐹^𝑘_{𝑝,𝑛+𝑘𝑝+1+1}𝑥,𝑦. (3.7) Also 𝑟₀=𝑟₁=⋯=𝑟_𝑝=0,𝑟_𝑛=𝑥𝑛+𝑘−𝑝−1𝑛−𝑝−1^{𝑛−𝑝−1}𝑦^𝑘+1. (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 𝑆^𝑘_𝑝(𝑥,𝑦;𝑡) of sequence {𝑠_𝑛} is  𝑆^𝑘_𝑝𝑥,𝑦;𝑡=𝐹^𝑘_{𝑝,𝑘𝑝+1+1}𝑥,𝑦+^𝑝_𝑖=1𝑡^𝑖𝐹^𝑘_{𝑝,𝑘𝑝+1+1+𝑖}𝑥,𝑦−𝑥𝐹^𝑘_{𝑝,𝑘𝑝+1+𝑖}+𝑦𝑥,𝑦_𝑘+1𝑡^𝑝+1(1−𝑥𝑡)^𝑘+11−𝑥𝑡−𝑦𝑡^𝑝+1−1. (3.9) Therefore, 𝑅^𝑘_𝑝𝑥,𝑦;𝑡=𝑡^{𝑘(𝑝+1)+1}𝑆^𝑘_𝑝𝑥,𝑦;𝑡. (3.10)

Theorem 3.3. The generating function of the incomplete bivariate Lucas 𝑝-polynomials is 𝑊^𝑘_𝑝𝑥,𝑦;𝑡=𝑡^𝑘(𝑝+1)𝐿_{𝑝,𝑘(𝑝+1)}𝑥,𝑦+^𝑝_𝑖=1𝑡^𝑖𝐿_{𝑝,𝑘(𝑝+1)+𝑖}𝑥,𝑦−𝑥𝐿_{𝑝,𝑘(𝑝+1)+𝑖−1}+𝑡𝑥,𝑦_𝑝+1𝑦^𝑘+1𝑝(1−𝑥𝑡)+1(1−𝑥𝑡)^𝑘+11−𝑥𝑡−𝑦𝑡^𝑝+1−1. (3.11)

Proof. From (2.9) and (3.3), 𝑊^𝑘_𝑝𝑥,𝑦;𝑡=^∞_𝑛=0𝐿^𝑘_𝑝,𝑛𝑥,𝑦𝑡^𝑛=^∞_𝑛=0𝐹^𝑘_𝑝,𝑛+1𝑥,𝑦+𝑝𝑦𝐹^𝑘−1_{𝑝,𝑛−𝑝}𝑥,𝑦𝑡^𝑛=^∞_𝑛=0𝐹^𝑘_𝑝,𝑛+1𝑥,𝑦𝑡^𝑛+𝑝𝑦^∞_𝑛=0𝐹^𝑘−1_{𝑝,𝑛−𝑝}𝑥,𝑦𝑡^𝑛=𝑡⁻¹𝑅^𝑘_𝑝𝑥,𝑦;𝑡+𝑝𝑦𝑡^𝑝𝑅_𝑝^𝑘−1𝑥,𝑦;𝑡. (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.