Incomplete 𝑘-Fibonacci and 𝑘-Lucas Numbers

Abstract

We define the incomplete k-Fibonacci and k-Lucas 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 𝐹_𝑛 are defined by the recurrence relation 𝐹₀=0,𝐹₁=1,𝐹_𝑛+1=𝐹_𝑛+𝐹_𝑛−1,𝑛⩾1. (1)

There exist a lot of properties about Fibonacci numbers. In particular, there is a beautiful combinatorial identity to Fibonacci numbers [1] 𝐹_𝑛=^{⌊(𝑛−1)/2⌋}_𝑖=0𝑖.𝑛−𝑖−1 (2) From (2), Filipponi [2] introduced the incomplete Fibonacci numbers 𝐹_𝑛(𝑠) and the incomplete Lucas numbers 𝐿_𝑛(𝑠). They are defined by 𝐹_𝑛(𝑠)=^𝑠_𝑗=0𝑗𝑛−1−𝑗𝑛=1,2,3,…;0≤𝑠≤𝑛−12,𝐿_𝑛(𝑠)=^𝑠_𝑗=0𝑛𝑗𝑛𝑛−𝑗𝑛−𝑗𝑛=1,2,3,…;0≤𝑠≤2. (3)

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 Jacobsthal-Lucas numbers. In [6], the authors define the incomplete Fibonacci and Lucas 𝑝-numbers. Also the authors define the incomplete bivariate Fibonacci and Lucas 𝑝-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 𝑘-Fibonacci Numbers.

For any positive real number 𝑘, the 𝑘-Fibonacci sequence, say {𝐹_𝑘,𝑛}_𝑛∈ℕ, is defined recurrently by 𝐹_𝑘,0=0,𝐹_𝑘,1=1,𝐹_𝑘,𝑛+1=𝑘𝐹_𝑘,𝑛+𝐹_{𝑘,𝑛−1},𝑛⩾1. (4)

In [8], 𝑘-Fibonacci numbers were found by studying the recursive application of two geometrical transformations used in the four-triangle longest-edge (4TLE) partition. These numbers have been studied in several papers; see [8–14].

For any positive real number 𝑘, the 𝑘-Lucas sequence, say {𝐿_𝑘,𝑛}_𝑛∈ℕ, is defined recurrently by 𝐿_𝑘,0=2,𝐿_𝑘,1=𝑘,𝐿_𝑘,𝑛+1=𝑘𝐿_𝑘,𝑛+𝐿_{𝑘,𝑛−1}. (5)

If 𝑘=1, we have the classical Lucas numbers. Moreover, 𝐿_𝑘,𝑛=𝐹_{𝑘,𝑛−1}+𝐹_𝑘,𝑛+1,𝑛⩾1; see [15].

In [12], the explicit formula to 𝑘-Fibonacci numbers is 𝐹_𝑘,𝑛=^{⌊(𝑛−1)/2⌋}_𝑖=0𝑖𝑘𝑛−𝑖−1_{𝑛−2𝑖−1}, (6) and the explicit formula of 𝑘-Lucas numbers is 𝐿_𝑘,𝑛(𝑥)=^⌊𝑛/2⌋_𝑖=0𝑛𝑖𝑘𝑛−𝑖𝑛−𝑖_𝑛−2𝑖. (7)

From (6) and (7), we introduce the incomplete 𝑘-Fibonacci and 𝑘-Lucas numbers and we obtain new recurrent relations, new identities, and their generating functions.

2. The Incomplete 𝑘-Fibonacci Numbers

Definition 1. The incomplete 𝑘-Fibonacci numbers are defined by 𝐹^𝑙_𝑘,𝑛=^𝑙_𝑖=0𝑖𝑘𝑛−1−𝑖_{𝑛−2𝑖−1},0≤𝑙≤𝑛−12. (8)

In Table 1, some values of incomplete 𝑘-Fibonacci numbers are provided.

Table 1 

The numbers 𝐹^𝑙_𝑘,𝑛, for 1⩽𝑛⩽10.

We note that 𝐹^{⌊(𝑛−1)/2⌋}_1,𝑛=𝐹_𝑛. (9) For 𝑘=1, we get incomplete Fibonacci numbers [2].

Some special cases of (8) are 𝐹⁰_𝑘,𝑛=𝑘^𝑛−1;(),𝐹𝑛≥1_1𝑘,𝑛=𝑘^𝑛−1+(𝑛−2)𝑘^𝑛−3;()𝐹𝑛≥3_2𝑘,𝑛=𝑘^𝑛−1+(𝑛−2)𝑘^𝑛−3+(𝑛−4)(𝑛−3)2𝑘^𝑛−5;()𝐹𝑛≥5_{⌊(𝑛−1)/2⌋𝑘,𝑛}=𝐹_𝑘,𝑛;()𝐹𝑛≥1_{⌊(𝑛−3)/2⌋𝑘,𝑛}=𝐹_𝑘,𝑛−𝑛𝑘2(𝑛even)𝐹_𝑘,𝑛−1(𝑛odd)(𝑛≥3). (10)

2.1. Some Recurrence Properties of the Numbers 𝐹^𝑙_𝑘,𝑛

Proposition 2. The recurrence relation of the incomplete 𝑘-Fibonacci numbers 𝐹^𝑙_𝑘,𝑛 is 𝐹^𝑙+1_𝑘,𝑛+2=𝑘𝐹^𝑙+1_𝑘,𝑛+1+𝐹^𝑙_𝑘,𝑛,0≤𝑙≤𝑛−22. (11)
The relation (11) can be transformed into the nonhomogeneous recurrence relation 𝐹^𝑙_𝑘,𝑛+2=𝑘𝐹^𝑙_𝑘,𝑛+1+𝐹^𝑙_𝑘,𝑛−𝑙𝑘𝑛−1−𝑙_{𝑛−1−2𝑙}. (12)

Proof. Use Definition 1 to rewrite the right-hand side of (11) as 𝑘^𝑙+1_𝑖=0𝑖𝑘𝑛−𝑖_𝑛−2𝑖+^𝑙_𝑖=0𝑖𝑘𝑛−𝑖−1_{𝑛−2𝑖−1}=^𝑙+1_𝑖=0𝑖𝑘𝑛−𝑖_{𝑛−2𝑖+1}+^𝑙+1_𝑖=1𝑘𝑛−𝑖𝑖−1^{𝑛−2𝑖+1}=𝑘^{𝑛−2𝑖+1𝑙+1}_𝑖=0𝑖+𝑛−𝑖𝑛−𝑖𝑖−1−𝑘^𝑛+1𝑛=−1_𝑙+1𝑖=0𝑖𝑘𝑛−𝑖+1_{𝑛−2𝑖+1}−0=𝐹^𝑙_𝑘,𝑛+2. (13)

Proposition 3. One has ^𝑠_𝑖=0𝑠𝑖𝐹^𝑙+𝑖_{𝑘,𝑛+𝑖}𝑘^𝑖=𝐹^𝑙+𝑠_{𝑘,𝑛+2𝑠},0≤𝑙≤𝑛−𝑠−12. (14)

Proof (by induction on 𝑠). Sum (14) clearly holds for 𝑠=0 and 𝑠=1 (see (11)). Now suppose that the result is true for all 𝑗<𝑠+1; we prove it for 𝑠+1: ^𝑠+1𝑖=0𝑖𝐹𝑠+1_{𝑙+𝑖𝑘,𝑛+𝑖}𝑘^𝑖=^𝑠+1_𝑖=0𝑠𝑖+𝑠𝐹𝑖−1^{𝑙+𝑖𝑘,𝑛+𝑖}𝑘^𝑖=^𝑠+1_𝑖=0𝑠𝑖𝐹^𝑙+𝑖_{𝑘,𝑛+𝑖}𝑘^𝑖+^𝑠+1_𝑖=0𝑠𝐹𝑖−1^{𝑙+𝑖𝑘,𝑛+𝑖}𝑘^𝑖=𝐹^𝑙+𝑠_{𝑘,𝑛+2𝑠}+𝑠𝐹𝑠+1^{𝑙+𝑠+1𝑘,𝑛+𝑠+1}𝑘^𝑠+1+^𝑠_𝑖=−1𝑠𝑖𝐹^𝑙+𝑖+1_{𝑘,𝑛+𝑖+1}𝑘^𝑖+1=𝐹^𝑙+𝑠_{𝑘,𝑛+2𝑠}+0+^𝑠_𝑖=0𝑠𝑖𝐹^𝑙+𝑖+1_{𝑘,𝑛+𝑖+1}𝑘^𝑖+1+𝑠𝐹−1^{𝑙𝑘,𝑛}=𝐹^𝑙+𝑠_{𝑘,𝑛+2𝑠}+𝑘^𝑠_𝑖=0𝑠𝑖𝐹^𝑙+𝑖+1_{𝑘,𝑛+𝑖+1}𝑘^𝑖+0=𝐹^𝑙+𝑠_{𝑘,𝑛+2𝑠}+𝑘𝐹^𝑙+𝑠+1_{𝑘,𝑛+2𝑠+1}=𝐹^𝑙+𝑠+1_{𝑘,𝑛+2𝑠+2}. (15)

Proposition 4. For 𝑛≥2𝑙+2, ^𝑠−1_𝑖=0𝐹^𝑙_{𝑘,𝑛+𝑖}𝑘^{𝑠−1−𝑖}=𝐹^𝑙+1_{𝑘,𝑛+𝑠+1}−𝑘^𝑠𝐹^𝑙+1_𝑘,𝑛+1. (16)

Proof (by induction on 𝑠). Sum (16) clearly holds for 𝑠=1 (see (11)). Now suppose that the result is true for all 𝑗<𝑠. We prove it for 𝑠: ^𝑠_𝑖=0𝐹^𝑙_{𝑘,𝑛+𝑖}𝑘^𝑠−𝑖=𝑘^𝑠−1_𝑖=0𝐹^𝑙_{𝑘,𝑛+𝑖}𝑘^{𝑠−𝑖−1}+𝐹^𝑙_{𝑘,𝑛+𝑠}=𝑘𝐹^𝑙+1_{𝑘,𝑛+𝑠+1}−𝑘^𝑠𝐹^𝑙+1_𝑘,𝑛+1+𝐹^𝑙_{𝑘,𝑛+𝑠}=𝑘𝐹^𝑙+1_{𝑘,𝑛+𝑠+1}+𝐹^𝑙_{𝑘,𝑛+𝑠}−𝑘^𝑠+1𝐹^𝑙+1_𝑘,𝑛+1=𝐹^𝑙+1_{𝑘,𝑛+𝑠+2}−𝑘^𝑠+1𝐹^𝑙+1_𝑘,𝑛+1. (17)

Note that if 𝑘, in (4), is a real variable, then 𝐹_𝑘,𝑛=𝐹_𝑥,𝑛 and they correspond to the Fibonacci polynomials defined by 𝐹_𝑛+1(𝑥)=1if𝑥𝑛=0if𝑛=1𝑥𝐹_𝑛(𝑥)+𝐹_𝑛−1(𝑥)if𝑛>1. (18)

Lemma 5. One has 𝐹^′_𝑛(𝑥)=𝑛𝐹_𝑛+1(𝑥)−𝑥𝐹_𝑛(𝑥)+𝑛𝐹_𝑛−1(𝑥)𝑥²=+4𝑛𝐿_𝑛(𝑥)−𝑥𝐹_𝑛(𝑥)𝑥².+4 (19)

See Proposition 13 of [12].

Lemma 6. One has ^{⌊(𝑛−1)/2⌋}_𝑖=0𝑖𝑖𝑘𝑛−1−𝑖_{𝑛−1−2𝑖}=𝑘²+4𝑛−4𝐹_𝑘,𝑛−𝑛𝑘𝐿_𝑘,𝑛2𝑘².+4 (20)

Proof. From (6) we have that 𝑘𝐹_𝑘,𝑛=^{⌊(𝑛−1)/2⌋}_𝑖=0𝑖𝑘𝑛−1−𝑖_𝑛−2𝑖. (21)
By deriving into the previous equation (respect to 𝑘), it is obtained 𝐹_𝑘,𝑛+𝑘𝐹^′_𝑘,𝑛=^{⌊(𝑛−1)/2⌋}_𝑖=0()𝑖𝑛−2𝑖𝑘𝑛−1−𝑖_{𝑛−2𝑖−1}=𝑛𝐹_𝑘,𝑛−2^{⌊(𝑛−1)/2⌋}_𝑖=0𝑖𝑖𝑘𝑛−1−𝑖_{𝑛−2𝑖−1}. (22)
From Lemma 5, 𝐹_𝑘,𝑛+𝑘𝑛𝐿_𝑘,𝑛−𝑘𝐹_𝑘,𝑛𝑘²+4=𝑛𝐹_𝑘,𝑛−2^{⌊(𝑛−1)/2⌋}_𝑖=0𝑖𝑖𝑘𝑛−1−𝑖_{𝑛−2𝑖−1}. (23)
From where, after some algebra (20) is obtained.

Proposition 7. One has ^{⌊(𝑛−1)/2⌋}_𝑙=0𝐹^𝑙_𝑘,𝑛=4𝐹_𝑘,𝑛+𝑛𝑘𝐿_𝑘,𝑛2𝑘²()𝑘+4𝑛𝑒𝑣𝑒𝑛₂+8𝐹_𝑘,𝑛+𝑛𝑘𝐿_𝑘,𝑛2𝑘²().+4𝑛𝑜𝑑𝑑 (24)

Proof. ^{⌊(𝑛−1)/2⌋}_𝑙=0𝐹^𝑙_𝑘,𝑛=𝐹⁰_𝑘,𝑛+𝐹¹_𝑘,𝑛+⋯+𝐹^{⌊(𝑛−1)/2⌋}_𝑘,𝑛=0𝑘𝑛−1−0_𝑛−1+0𝑘𝑛−1−0_𝑛−1+1𝑘𝑛−1−1_𝑛−30+⋯+𝑘𝑛−1−0_𝑛−1+1𝑘𝑛−1−1_𝑛−3+⋯+𝑛−1−𝑛−12𝑛−12𝑘^{𝑛−1−2⌊(𝑛−1)/2⌋}=𝑛−120+1𝑘𝑛−1−0_𝑛−1+𝑛−121𝑘𝑛−1−1_𝑛−3+⋯+𝑛−1−𝑛−12𝑛−12𝑘^{𝑛−1−2⌊(𝑛−1)/2⌋}=^{⌊(𝑛−1)/2⌋}_𝑖=0𝑛−12𝑖+1−𝑖𝑘𝑛−1−𝑖_{𝑛−1−2𝑖}=^{⌊(𝑛−1)/2⌋}_𝑖=0𝑛−12𝑖+1𝑘𝑛−1−𝑖_{𝑛−1−2𝑖}−^{⌊(𝑛−1)/2⌋}_𝑖=0𝑖𝑖𝑘𝑛−1−𝑖_{𝑛−1−2𝑖}=𝑛−12+1𝐹_𝑘,𝑛−^{⌊(𝑛−1)/2⌋}_𝑖=0𝑖𝑖𝑘𝑛−1−𝑖_{𝑛−1−2𝑖}. (25) From Lemma 6, (24) is obtained.

3. The Incomplete 𝑘-Lucas Numbers

Definition 8. The incomplete 𝑘-Lucas numbers are defined by 𝐿^𝑙_𝑘,𝑛=^𝑙_𝑖=0𝑛𝑖𝑘𝑛−𝑖𝑛−𝑖_𝑛−2𝑖𝑛,0≤𝑙≤2. (26)

In Table 2, some numbers of incomplete 𝑘-Lucas numbers are provided.

Table 2 

The numbers 𝐿^𝑙_𝑘,𝑛, for 1⩽𝑛⩽9.

We note that 𝐿^⌊𝑛/2⌋_1,𝑛=𝐿_𝑛. (27)

Some special cases of (26) are 𝐿⁰_𝑘,𝑛=𝑘^𝑛;(),𝐿𝑛≥1_1𝑘,𝑛=𝑘^𝑛+𝑛𝑘^𝑛−2;(),𝐿𝑛≥2_2𝑘,𝑛=𝑘^𝑛+𝑛𝑘^𝑛−2+𝑛(𝑛−3)2𝑘^𝑛−4;(),𝐿𝑛≥4_{⌊𝑛/2⌋𝑘,𝑛}=𝐿_𝑘,𝑛;(),𝐿𝑛≥1_{⌊(𝑛−2)/2⌋𝑘,𝑛}=𝐿_𝑘,𝑛−2(𝑛even)𝐿_𝑘,𝑛−𝑛𝑘(𝑛odd)(𝑛≥2). (28)

3.1. Some Recurrence Properties of the Numbers 𝐿^𝑙_𝑘,𝑛

Proposition 9. One has 𝐿^𝑙_𝑘,𝑛=𝐹^𝑙−1_{𝑘,𝑛−1}+𝐹^𝑙_𝑘,𝑛+1𝑛,0≤𝑙≤2. (29)

Proof. By (8), rewrite the right-hand side of (29) as ^{𝑙−1𝑖=0}𝑖𝑘𝑛−2−𝑖_{𝑛−2−2𝑖}+^𝑙_𝑖=0𝑖𝑘𝑛−𝑖_𝑛−2𝑖=^𝑙_𝑖=1𝑘𝑛−1−𝑖𝑖−1^𝑛−2𝑖+^𝑙_𝑖=0𝑖𝑘𝑛−𝑖_𝑛−2𝑖=^𝑙_𝑖=0+𝑖𝑘𝑛−1−𝑖𝑖−1𝑛−𝑖_𝑛−2𝑖−=𝑛−1−1_𝑙𝑖=0𝑛𝑖𝑘𝑛−𝑖𝑛−𝑖_𝑛−2𝑖+0=𝐿^𝑙_𝑘,𝑛. (30)

Proposition 10. The recurrence relation of the incomplete 𝑘-Lucas numbers 𝐿^𝑙_𝑘,𝑛 is 𝐿^𝑙+1_𝑘,𝑛+2=𝑘𝐿^𝑙+1_𝑘,𝑛+1+𝐿^𝑙_𝑘,𝑛𝑛,0≤𝑙≤2. (31)
The relation (31) can be transformed into the nonhomogeneous recurrence relation 𝐿^𝑙_𝑘,𝑛+2=𝑘𝐿^𝑙_𝑘,𝑛+1+𝐿^𝑙_𝑘,𝑛−𝑛𝑙𝑘𝑛−𝑙𝑛−𝑙_𝑛−2𝑙. (32)

Proof. Using (29) and (11), we write 𝐿^𝑙+1_𝑘,𝑛+2=𝐹^𝑙_𝑘,𝑛+1+𝐹^𝑙+1_𝑘,𝑛+3=𝑘𝐹^𝑙_𝑘,𝑛+𝐹^𝑙−1_{𝑘,𝑛−1}+𝑘𝐹^𝑙+1_𝑘,𝑛+2+𝐹^𝑙_𝑘,𝑛+1=𝑘𝐹^𝑙_𝑘,𝑛+𝐹^𝑙+1_𝑘,𝑛+2+𝐹^𝑙−1_{𝑘,𝑛−1}+𝐹^𝑙_𝑘,𝑛+1=𝑘𝐿^𝑙+1_𝑘,𝑛+1+𝐿^𝑙_𝑘,𝑛. (33)

Proposition 11. One has 𝑘𝐿^𝑙_𝑘,𝑛=𝐹^𝑙_𝑘,𝑛+2−𝐹^𝑙−2_{𝑘,𝑛−2},0≤𝑙≤𝑛−12. (34)

Proof. By (29), 𝐹^𝑙_𝑘,𝑛+2=𝐿^𝑙_𝑘,𝑛+1−𝐹^𝑙−1_𝑘,𝑛,𝐹^𝑙−2_{𝑘,𝑛−2}=𝐿^𝑙−1_{𝑘,𝑛−1}−𝐹^𝑙−1_𝑘,𝑛, (35) whence, from (31), 𝐹^𝑙_𝑘,𝑛+2−𝐹^𝑙−2_{𝑘,𝑛−2}=𝐿^𝑙_𝑘,𝑛+1−𝐿^𝑙−1_{𝑘,𝑛−1}=𝑘𝐿^𝑙_𝑘,𝑛. (36)

Proposition 12. One has ^𝑠_𝑖=0𝑠𝑖𝐿^𝑙+𝑖_{𝑘,𝑛+𝑖}𝑘^𝑖=𝐿^𝑙+𝑠_{𝑘,𝑛+2𝑠},0≤𝑙≤𝑛−𝑠2. (37)

Proof. Using (29) and (14), we write ^𝑠_𝑖=0𝑠𝑖𝐿^𝑙+𝑖_{𝑘,𝑛+𝑖}𝑘^𝑖=^𝑠_𝑖=0𝑠𝑖𝐹^{𝑙+𝑖−1}_{𝑘,𝑛+𝑖−1}+𝐹^𝑙+𝑖_{𝑘,𝑛+𝑖+1}𝑘^𝑖=^𝑠_𝑖=0𝑠𝑖𝐹^{𝑙+𝑖−1}_{𝑘,𝑛+𝑖−1}𝑘^𝑖+^𝑠_𝑖=0𝑠𝑖𝐹^𝑙+𝑖_{𝑘,𝑛+𝑖+1}𝑘^𝑖=𝐹^{𝑙−1+𝑠}_{𝑘,𝑛−1+2𝑠}+𝐹^𝑙+𝑠_{𝑘,𝑛+1+2𝑠}=𝐿^𝑙+𝑠_{𝑘,𝑛+2𝑠}. (38)

Proposition 13. For 𝑛≥2𝑙+1, ^𝑠−1_𝑖=0𝐿^𝑙_{𝑘,𝑛+𝑖}𝑘^{𝑠−1−𝑖}=𝐿^𝑙+1_{𝑘,𝑛+𝑠+1}−𝑘^𝑠𝐿^𝑙+1_𝑘,𝑛+1. (39)

The proof can be done by using (31) and induction on 𝑠.

Lemma 14. One has ^⌊𝑛/2⌋_𝑖=0𝑖𝑛𝑖𝑘𝑛−𝑖𝑛−𝑖_𝑛−2𝑖=𝑛2𝐿_𝑘,𝑛−𝑘𝐹_𝑘,𝑛. (40)

The proof is similar to Lemma 6.

Proposition 15. One has ^⌊𝑛/2⌋_𝑙=0𝐿^𝑙_𝑘,𝑛=𝐿_𝑘,𝑛+𝑛𝑘2𝐹_𝑘,𝑛()1𝑛𝑒𝑣𝑒𝑛2𝐿_𝑘,𝑛+𝑛𝑘𝐹_𝑘,𝑛(𝑛𝑜𝑑𝑑). (41)

Proof. An argument analogous to that of the proof of Proposition 7 yields ^⌊𝑛/2⌋_𝑙=0𝐿^𝑙_𝑘,𝑛=𝑛2+1𝐿_𝑘,𝑛−^⌊𝑛/2⌋_𝑖=0𝑖𝑛𝑖𝑘𝑛−𝑖𝑛−𝑖_𝑛−2𝑖. (42)
From Lemma 14, (41) is obtained.

4. Generating Functions of the Incomplete 𝑘-Fibonacci and 𝑘-Lucas Number

In this section, we give the generating functions of incomplete 𝑘-Fibonacci and 𝑘-Lucas numbers.

Lemma 16 (see [3, page 592]). Let {𝑠_𝑛}^∞_𝑛=0 be a complex sequence satisfying the following nonhomogeneous recurrence relation: 𝑠_𝑛=𝑎𝑠_𝑛−1+𝑏𝑠_𝑛−2+𝑟_𝑛(𝑛>1), (43) where 𝑎 and 𝑏 are complex numbers and {𝑟_𝑛} is a given complex sequence. Then, the generating function 𝑈(𝑡) of the sequence {𝑠_𝑛} is 𝑈(𝑡)=𝐺(𝑡)+𝑠₀−𝑟₀+𝑠₁−𝑠₀𝑎−𝑟₁𝑡1−𝑎𝑡−𝑏𝑡², (44) where 𝐺(𝑡) denotes the generating function of {𝑟_𝑛}.

Theorem 17. The generating function of the incomplete 𝑘-Fibonacci numbers 𝐹^𝑙_𝑘,𝑛 is given by 𝑅_𝑘,𝑙(𝑥)=^∞_𝑖=0𝐹^𝑙_𝑘,𝑖𝑡^𝑖=𝑡^2𝑙+1𝐹_𝑘,2𝑙+1+𝐹_𝑘,2𝑙+2−𝑘𝐹_𝑘,2𝑙+1𝑡𝑡−²(1−𝑘𝑡)^𝑙+1×1−𝑘𝑡−𝑡²⁻¹. (45)

Proof. Let 𝑙 be a fixed positive integer. From (8) and (12), 𝐹^𝑙_𝑘,𝑛=0 for 0≤𝑛<2𝑙+1, 𝐹^𝑙_𝑘,2𝑙+1=𝐹_𝑘,2𝑙+1, and 𝐹^𝑙_𝑘,2𝑙+2=𝐹_𝑘,2𝑙+2, and 𝐹^𝑙_𝑘,𝑛=𝑘𝐹^𝑙_{𝑘,𝑛−1}+𝐹^𝑙_{𝑘,𝑛−2}−𝑙𝑘𝑛−3−𝑙_{𝑛−3−2𝑙}. (46)
Now let 𝑠₀=𝐹^𝑙_𝑘,2𝑙+1,𝑠₁=𝐹^𝑙_𝑘,2𝑙+2,𝑠_𝑛=𝐹^𝑙_{𝑘,𝑛+2𝑙+1}. (47)
Also let 𝑟₀=𝑟₁=0,𝑟_𝑛=𝑘𝑛+𝑙−2𝑛−2^𝑛−2. (48) The generating function of the sequence {𝑟_𝑛} is 𝐺(𝑡)=𝑡²/(1−𝑘𝑡)^𝑙+1 (see [16, page 355]). Thus, from Lemma 16, we get the generating function 𝑅_𝑘,𝑙(𝑥) of sequence {𝑠_𝑛}.

Theorem 18. The generating function of the incomplete 𝑘-Lucas numbers 𝐿^𝑙_𝑘,𝑛 is given by 𝑆_𝑘,𝑙(𝑥)=^∞_𝑖=0𝐿^𝑙_𝑘,𝑖𝑡^𝑖=𝑡^2𝑙𝐿_𝑘,2𝑙+𝐿_𝑘,2𝑙+1−𝑘𝐿_𝑘,2𝑙𝑡𝑡−²(2−𝑡)(1−𝑘𝑡)^𝑙+1×1−𝑘𝑡−𝑡²⁻¹. (49)

Proof. The proof of this theorem is similar to the proof of Theorem 17. Let 𝑙 be a fixed positive integer. From (26) and (32), 𝐿^𝑙_𝑘,𝑛=0 for 0≤𝑛<2𝑙, 𝐿^𝑙_𝑘,2𝑙=𝐿_𝑘,2𝑙, and 𝐿^𝑙_𝑘,2𝑙+1=𝐿_𝑘,2𝑙+1, and 𝐿^𝑙_𝑘,𝑛=𝑘𝐿^𝑙_{𝑘,𝑛−1}+𝐿^𝑙_{𝑘,𝑛−2}−𝑛−2𝑘𝑛−2−𝑙𝑛−2−𝑙𝑛−2−2𝑙^{𝑛−2−2𝑙}. (50)
Now let 𝑠₀=𝐿^𝑙_𝑘,2𝑙,𝑠₁=𝐿^𝑙_𝑘,2𝑙+1,𝑠_𝑛=𝐿^𝑙_{𝑘,𝑛+2𝑙}. (51)
Also let 𝑟₀=𝑟₁=0,𝑟_𝑛=𝑘𝑛+2𝑙−2𝑛+𝑙−2^{𝑛+2𝑙−2}. (52) The generating function of the sequence {𝑟_𝑛} is 𝐺(𝑡)=𝑡²(2−𝑡)/(1−𝑘𝑡)^𝑙+1 (see [16, page 355]). Thus, from Lemma 16, we get the generating function 𝑆_𝑘,𝑙(𝑥) of sequence {𝑠_𝑛}.

5. Conclusion

In this paper, we introduce incomplete 𝑘-Fibonacci and 𝑘-Lucas numbers, and we obtain new identities. In [17], the authors introduced the ℎ(𝑥)-Fibonacci polynomials. That generalizes Catalan’s Fibonacci polynomials and the 𝑘-Fibonacci numbers. Let ℎ(𝑥) be a polynomial with real coefficients. The ℎ(𝑥)-Fibonacci polynomials {𝐹_ℎ,𝑛(𝑥)}_𝑛∈ℕ are defined by the recurrence relation 𝐹_ℎ,0(𝑥)=0,𝐹_ℎ,1(𝑥)𝐹=1,_ℎ,𝑛+1(𝑥)=ℎ(𝑥)𝐹_ℎ,𝑛(𝑥)+𝐹_ℎ,𝑛−1(𝑥),𝑛⩾1. (53)

It would be interesting to study a definition of incomplete ℎ(𝑥)-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. USA-II-2011-0059.