Relation Between Lah Matrix and K-Fibonacci Matrix

Relation between Lah matrix and k-Fibonacci Matrix

Irda Melina Zet#1, Sri Gemawati#2, Kartini Kartini#3 #Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Riau Bina Widya Campus, Pekanbaru 28293, Indonesia

Abstract. - The Lah matrix is represented by 퐿푛, is a matrix where each entry is Lah number. Lah number is count the number of ways a set of n elements can be partitioned into k nonempty linearly ordered subsets. k-Fibonacci matrix, 퐹푛(푘) is a matrix which all the entries are k-Fibonacci numbers. k-Fibonacci numbers are consist of the first term being 0, the second term being 1 and the next term depends on a natural number k. In this paper, a new matrix is defined namely 퐴푛 where it is not commutative to multiplicity of two matrices, so that another matrix 퐵푛 is defined such that 퐴푛 ≠ 퐵푛. The result is two forms of factorization from those matrices. In addition, the properties of the relation of Lah matrix and k- Fibonacci matrix is yielded as well.

Keywords — Lah numbers, Lah matrix, k-Fibonacci numbers, k-Fibonacci matrix.

I. INTRODUCTION Guo and Qi [5] stated that Lah numbers were introduced in 1955 and discovered by Ivo. Lah numbers are count the number of ways a set of n elements can be partitioned into k non-empty linearly ordered subsets. Lah numbers [9] is denoted by 퐿(푛, 푘) for every 푛, 푘 are elements of integers with initial value 퐿(0,0) = 1. Lah numbers can also be presented in Lah matrix where all the entries are Lah numbers and denoted by 퐿푛 [9]. Reference [1] discussed about Stirling matrix and Lah matrix with the inverses and yields some interesting properties. Reference [5] gives six proofs of Lah numbers identity property by using Lah numbers generator function, Chu- Vandermonde addition formula, inverse formula, Gauss hypergeometric series, and first kind Stirling numbers generator function. In [9] discussed Lah numbers and Lindstorm theorem by giving a combinatorial interpretation from Lah numbers through planar network resulting in some properties. Moreover, k-Fibonacci numbers were introduced by Falcon [2] and denoted by 퐹푛,푘 where 푛, 푘 are elements of integers.The k-Fibonacci numbers can be represented into k-Fibonacci matrix where all entries are k-Fibonacci numbers. It is denoted by 퐹푛(푘). Reference [8] talks about the relation of k-Fibonacci sequence and its generalization. Wahyuni et al. [13] introduces some identities of k-Fibonacci sequence modulo ring. Reference [2] discusses the relation between k-Fibonacci matrix and Pascal matrix where two new matrices are yielded. [10] also talks about the relation of Bell polynomial matrix and k-Fibonacci matrix such that two new matrices as form of factorization of Bell polynomial matrix. This article discusses Lah matrix and k-Fibonacci matrix, invers of k-Fibonacci matrix which is used to find new matrices, and relationship of Lah matrix and k-Fibonacci matrix. The second section of this article discusses about Lah matrix and k-Fibonacci matrix. The third section contains the main result which is the relationship of Lah matrix and k- Fibonacci matrix. Conclusion is given in the last section.

II. LAH MATRIX AND K-FIBONACCI MATRIX In this section, we recall the notions of Lah and k-Fibonacci matrices and review some properties which we will need in the next section. Some definitions and theories related to Lah and k-Fibonacci matrices that have been discussed by several authors will also be presented. Lah numbers are defined by Guo and Qi [5] as follows. Definition 2.1. Lah numbers, 퐿(푛, 푘) are defined as 푛! 퐿(푛, 푘) = (푛−1) . 푘−1 푘! for 푛, 푘 are natural numbers. It can be concluded that If 푘 = 1 then 퐿(푛, 1) = 푛!. If 푘 = 푛 then 퐿(푛, 푛) = 1. If 푛 < 푘 then 퐿(푛, 푘) = 0. If 푘 = 0 then 퐿(푛, 0) = 0.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Irda Melina Zet et al. / IJMTT, 66(10), 116-122, 2020

Martinjak [9] gives recursive formula for Lah numbers: 퐿(푛 + 1, 푘) = 퐿(푛, 푘 − 1) + (푛 + 푘)퐿(푛, 푘). where 푛, 푘 are integers and 푛 ≥ 푘 with intial value 퐿(0,0) = 1. Lah matrix is a lower triangle matrix with every entry is Lah number. Definition of Lah matrix according to Martinjak [9] is given below.

Definition 2.2. Let 퐿(푖, 푗) be Lah numbers. The 푛 × 푛 Lah matrix is denoted by 퐿푛 = [푙푖,푗] where 푖, 푗 = 1, 2, … , 푛 and defined as 퐿(푖, 푗) if 푖 ≥ 푗, 푙 = { 푖,푗 0, otherwise. The general form of Lah matrix is 1 0 0 0 0 0

푙2,1 1 0 0 0 0 푙 푙3,1 3,2 1 0 0 0 퐿푛 = . 푙4,1 푙4,2 푙4,3 1 0 0 ⋮ ⋮ ⋮ ⋮ ⋱ 0 [ 푙푛,1 푙푛,푛−1 푙푛,푛−2 푙푛,푛−3 ⋯ 1]

From the general form above, it can be seen that the main diagonal is 1 and the determinant (det) of 퐿푛 is obtained from the multiplication of the diagonal entries and so det (퐿푛) = 1. Since det (퐿푛) ≠ 0 then 퐿푛 has inverse. Engbers et al. [1] defines matrix inverse of Lah matrix as follows.

−1 Definition 2.3. Let 퐿(푖, 푗) be Lah numbers and 퐿푛 = [푙푖,푗] be 푛 × 푛 Lah matrix. The inverse of Lah matrix is 퐿푛 = 푖−푗 [(−1) 푙푖,푗] where 푖, 푗 = 1, 2, … , 푛.

The following is the general form of Lah matrix: 1 0 0 0 0 0

−푙2,1 1 0 0 0 0 푙 −푙 −1 3,1 3,2 1 0 0 0 퐿푛 = . −푙4,1 푙4,2 −푙4,3 1 0 0 ⋮ ⋮ ⋮ ⋮ ⋱ 0 [ 푙푛,1 −푙푛,푛−1 푙푛,푛−2 −푙푛,푛−3 ⋯ 1]

The notion of k-Fibonacci numbers defined by Falcon [3] is given below.

th Definition 2.4. For any integer number 푘 ≥ 1, the 푘 Fibonacci sequence, say {퐹푘,푛}푛∈푁 is defined recurrently by 퐹푘,0 = 0, 퐹푘,1 = 1, dan 퐹푘,푛+1 = 푘 퐹푘,푛 + 퐹푘,푛−1 for any 푛 ≥ 1. The k-Fibonacci numbers can also be represented in a lower triangular matrix where each entry is k-Fibonacci number. This definition is given in [2].

푡ℎ Definition 2.5. Let 퐹푘,푛 be 푛 k-Fibonacci number, the 푛 × 푛 k-Fibonacci matrix as the unipotent lower triangular matrix Fn (푘) = [푓푖,푗]푖,푗=1,…,푛 defined with entries. That is 퐹 if 푖 ≥ 푗, 푓 (푘) = { 푘,푖−푗+1 푖,푗 0, otherwise 푖 < 푗. Generally, k-Fibonacci matrix is written in this form

1 0 0 0 0 0

퐹푘,2 1 0 0 0 0 퐹 퐹푘,3 푘,2 1 0 0 0 퐹푛(푘) = . 퐹푘,4 퐹푘,3 퐹푘,2 1 0 0 ⋮ ⋮ ⋮ ⋮ ⋱ 0 [ 퐹푘,푛 퐹푘,푛−1 퐹푘,푛−2 퐹푘,푛−3 ⋯ 1]

117 Irda Melina Zet et al. / IJMTT, 66(10), 116-122, 2020

From the above general formula of k-Fibonacci matrix it can be seen that the main diagonal is 1 and determinant (det) is yielded from the multiplication of the entries of the diagonal and so that det (퐹푛(푘)) = 1. Since det 퐹푛(푘)) ≠ 0, 퐹푛(푘) is has inverse. Falcon [2] defines the inverse matrix of the k-Fibonacci matrix as follows.

−1 Definition 2.6. Let F 푛(푘) be inverse matrix of the k-Fibonacci matrix, the 푛 × 푛 inverse k-Fibonacci matrix as lower −1 ′ triangular matrix F 푛(푘) = [푓푖,푗(푘)]푖,푗=1,…,푛 where 1 푖푓 푗 = 푖,

′ −푘 푖푓 푗 = 푖 − 1, 푓푖,푗(푘) = −1 푖푓 푗 = 푖 − 2, { 0 표푡ℎ푒푟푤푖푠푒,

−1 −1 Since 퐹푛(푘) has inverse, then 퐹푛(푘) 퐹푛 (푘) = 퐼푛 = 퐹푛 (푘)퐹푛(푘). Therefore k-Fibonacci 퐹푛(푘) is an invertible matrix. Generally k-Fibonacci matrix can be written by 1 0 0 0 0 0 −푘 1 0 0 0 0

퐹−1(푘) = −1 −푘 1 0 0 0 . 푛 0 −1 −푘 1 0 0 ⋮ ⋱ ⋱ ⋱ ⋱ 0 [ 0 0 0 −1 −푘 1]

From the general form above, it can be concluded that each entry of the inverse of the k-Fibonacci matrix is −1 1, −푘, −1, and 0. In other words, the entry of 퐹푛 (푘) for any 푛 is fixed.

III. RELATION BETWEEN LAH MATRIX AND K-FIBONACCI MATRIX Relationship of Lah matrix and k-Fibonacci matrix is obtained from multiplication of two matrices, which is multiplication of inverse of k-Fibonacci matrix and Lah matrix and the other way around. It starts from multiplying 2 × 2 matrices, follows by 3 × 3 and eventually a new matrix is obtained. To construct the general form of the new matrix, the properties of this new matrix should be investigated. Therefore, a larger size of matrices are needed. To make the calculation easier, software Maple 13 is used. In the first factorization, a new matrix 퐴푛, for any n integer, is obtained from multiplying inverse of k-Fibonacci matrix and Lah matrix. In the second factorization, a new matrix 퐵푛 is yielded from multiplying Lah matrix and k-Fibonacci inverse matrix.

A. First Factorization of Lah matrix and k-Fibonacci Matrix To get the relationship of Lah matrix and k-Fibonacci matrix, multiply two 2 × 2 matrices which are inverse of k- −1 Fibonacci matrix 퐹2 (푘) and 퐿2 Lah matrix and obtain 1 0 1 0 1 0 퐹−1(푘) 퐿 = [ ] [ ] = [ ] = 퐴 . 2 2 −푘 1 2 1 2 − 푘 1 2 −1 For 푛 = 3, by multiplying inverse of k-Fibonacci 퐹3 (푘) matrix and Lah matrix 퐿3 it is obtained 1 0 0 1 0 0 1 0 0 −1 퐹3 (푘) 퐿3 = [−푘 1 0] [2 1 0] = [ 2 − 푘 1 0] = 퐴3. −1 −푘 1 6 6 1 5 − 2푘 6 − 푘 1 Then, for 푛 > 3, multiplication of inverse of k-Fibonacci matrix and Lah matrix is done by Maple 13. By investigating each entry of 퐴4 for 푖 = 푗 entry the entries of the diagonal is 1 and for 푖 > 푗 the following construction is given :

i. In the first row, 푎1,1 = 1, 푎1,푗 = 0 for 푗 ≥ 2.

ii. In the second row, 푎2,1 = 2 − 푘, 푎2,2 = 1, 푎2,푗 = 0 for 푗 ≥ 3.

iii. In the third row 푎3,1 = 5 − 2푘, 푎3,2 = 6 − 푘, 푎3,3 = 1, 푎3,푗 = 0 for 푗 ≥ 3.

iv. 푎푖,푗 = 1 for any 푖 = 푗 and for 푖 < 푗, 푎푖,푗 = 0.

The entry of 퐴4 matrix is listed in the following Table 3.1 By investigating the entries of 퐴4 for 푖 ≥ 푗, the value of entries of 퐴4 can be derived as listed in the table below.

118 Irda Melina Zet et al. / IJMTT, 66(10), 116-122, 2020

Table 3.1: Elements of 퐴푛

Entries of 퐴푛 Value of the entries of 퐴푛

푎1,1 (1) 푙1,1 = 퐿(1,1)

푎2,2 (1) 푙2,2 = 퐿(2,2)

푎3,3 (1) 푙3,3 = 퐿(3,3)

푎4,4 (1) 푙4,4 = 퐿(4,4)

푎2,1 (1) 푙2,1 + (−푘) 푙1,1 = 퐿(2,1) − 푘퐿(1,1)

푎3,2 (1)푙3,2 + (−푘) 푙2,2 = 퐿(3,2) − 푘퐿(2,2)

푎4,3 (1)푙4,3 + (−푘) 푙3,3 = 퐿(4,3) − 푘퐿(3,3)

푎3,1 (1)푙3,1 + (−푘) 푙2,1 + (−1) 푙1,1 = 퐿(3,1) − 푘퐿(2,1) − 퐿(1,1)

푎4,2 (1)푙4,2 + (−푘) 푙3,2 + (−1) 푙2,2 = 퐿(4,2) − 푘퐿(3,2) − 퐿(2,2)

푎4,1 (1)푙4,1 + (−푘) 푙3,1 + (−1) 푙2,1 + (0) (1)푙1,1 = 퐿(4,1) − 푘퐿(3,1) − 퐿(2,1) ⋮ ⋮

푎푖,푗 퐿(푖, 푗) − 푘 퐿(푖 − 1, 푗) − 퐿(푖 − 2, 푗)

Hence, multiplication of inverse of k-Fibonacci matrix and Lah matrix yields a new matrix 퐴푛 and the definition generally is given as follows.

Definition 3.1. For every natural number 푛, it is defined an (푛 + 1) × (푛 + 1) matrix 퐴푛 = [푎푖,푗] with 푖, 푗 = 0,1,2, … , 푛 as follows 푎푖,푗 = 퐿(푖, 푗) − 푘 퐿(푖 − 1, 푗) − 퐿(푖 − 2, 푗).

From the definition above, 푎푖,푗 = 1 for every 푖 = 푗, 푎푖,푗 = 0 when 푖 < 푗. For every 푖 > 푗, apply 푎푖,푗 = 퐿(푖, 푗) − 푘 퐿(푖 − 1, 푗) − 퐿(푖 − 2, 푗).

From definitions of Lah matrix, k-Fibonacci matrix, and 퐴푛 matrix, the following theorem is derived. Theorem 3.2. Lah matrix defined in Definition 2.2 can be defined as multiplication of k-Fibonacci 퐹푛(푘) defined in Definition 2.5 with 퐴푛 and given as 퐿푛 = 퐹푛(푘) 퐴푛.

Proof. Since k-Fibonacci matrix 퐹푛(푘) has inverse, it will be proven that −1 퐴푛 = 퐹푛 (푘) 퐿푛.

−1 Suppose 퐹푛 (푘) is inverse of k-Fibonacci matrix so that the main diagonal is 1. 퐿푛 is Lah matrix and so the main −1 diagonal is also 1. Multiplication of 퐹푛 (푘) with 퐿푛 yields a new matrix with main diagonal 1. If 푖 = 푗 then 푎푖,푗 = 1 and if 푖 < 푗 then 푎푖,푗 = 0, and for every 푖 > 2 then ′ ′ ′ ′ ′ 푎푖,푗 = 푓푖,푖(푘) 푙푖,푗 + 푓푖,푖−1(푘) 푙푖−1,푗 + 푓푖,푖−2(푘) 푙푖−2,푗 + 푓푖,푖−3(푘) 푙푖−3,푗 + ⋯ + 푓푖,푛(푘) 푙푛,푗, 푛 ′ = ∑ 푓푖,푟(푘) 푙푟,푗. 푟=1 −1 It can be concluded that 퐹푛 (푘) 퐿푛 = 퐴푛. Hence 퐿푛 = 퐹푛(푘) 퐴푛. ∎ Theorem 3.2 gives the relation of Lah numbers and k-Fibonacci numbers defined in Theorem 3.3. Theorem 3.3. Suppose 퐿(푖, 푗) is Lah number defined in Definition 2.1 and 퐹푘,푛 is the k-Fibonacci numbers defined in Definition 2.5. Then for 푖 ≥ 푗 + 2, 푖 2 퐿(푖, 푗) = 퐹푘,푖−푗+1 + (푗 + 푗 − 푘) 퐹푘,푖−푗 + ∑ (퐿(푟, 푗) − 푘 퐿(푟 − 1, 푗) − 퐿(푟 − 2, 푗))퐹푘,푖−푟+1. 푟=푗+2 2 For 푖 < 푗 + 2 it is obtained that 퐿(푖, 푗) = 퐹푘,푖−푗+1 + (푗 + 푗 − 푘) 퐹푘,푖−푗 and for 푖 < 푗 then 퐿(푖, 푗) = 0 for 푖 to be natural numbers.

Proof. From Definition 3.2 푎푗,푗 = 퐿(푗, 푗) = 1, 푎푗+1,푗 = 퐿(푗 + 1, 푗) − 푘퐿(푗, 푗) − 퐿(푗 − 1, 푗) = 푗(푗 + 1) − 푘(1) − 0 = 푗2 + 푗 − 푘,

119 Irda Melina Zet et al. / IJMTT, 66(10), 116-122, 2020

for 푟 ≥ 푗 + 2 푎푡,푗 = 퐿(푟, 푗) − 푘퐿(푟 − 1, 푗) − 퐿(푟 − 2, 푗).

Definition 2.5 and Theorem 3.2 gives the following 퐿(푖, 푗) = 푙푖,푗 푖

= ∑ 푓푖,푟(푘) 푎푟,푗 푟=푗 푖

= ∑ 퐹푘,푖−푟+1 푎푟,푗 푟=푗 푖

= 퐹푘,푖−푗+1 푎푗,푗 + 퐹푘,푖−(푗+1)+1 푎푗+1,푗 + ∑ 퐹푘,푖−푟+1 푎푟,푗 푟=푗+2 푖 2 = 퐹푘,푖−푗+1 + 퐹푘,푖−푗(푗 + 푗 − 푘) + ∑ 퐹푘,푖−푟+1 (퐿(푟, 푗) − 푘퐿(푟 − 1, 푗) − 퐿(푟 − 2, 푗)) 푟=푗+2 2 푖 퐿(푖, 푗) = 퐹푘,푖−푗+1 + (푗 + 푗 − 푘) 퐹푘,푖−푗 + ∑푟=푗+2(퐿(푟, 푗) − 푘퐿(푟 − 1, 푗) − 퐿(푟 − 2, 푗)) 퐹푘,푖−푟+1. ∎

B. Second Factorization for Lah matrix and k-Fibonacci Matrix To find the relation of Lah matrix and k-Fibonacci matrix from second factorization, multiplication of two matrices is −1 needed. It is done by multiplying Lah matrix 퐿푛 and 퐹푛 (푘) inverse of k-Fibonacci matrix and eventually yielding a new matrix called 퐵푛. It starts from multiplying 2 × 2 matrices such that the following is obtained. 1 0 1 0 1 0 퐿 퐹−1(푘) = [ ] [ ] = [ ] = 퐵 . 2 2 2 1 −푘 1 2 − 푘 1 2 Then proceed with multiplication of 3 × 3 matrices to get 1 0 0 1 0 0 1 0 0 −1 퐿3 퐹3 (푘) = [2 1 0] [−푘 1 0] = [ 2 − 푘 1 0] = 퐵3. 6 6 1 −1 −푘 1 5 − 6푘 6 − 푘 1 Then, for 푛 > 3 is done by Maple 13 to get the general form. By observing each entry of 퐵4 it can be seen that for 푖 = 푗 the entry in the main diagonal is 1 and for 푖 > 푗 a simple construction is obtained as follows:

i. In the first row 푏1,1 = 1, 푏1,푗 = 0for 푗 ≥ 2.

ii. In the second row 푏2,1 = 2 − 푘, 푏2,2 = 1, 푏2,푗 = 0 for 푗 ≥ 3.

iii. In the third row 푏3,1 = 5 − 6푘, 푏3,2 = 6 − 푘, 푏3,3 = 1, 푏3,푗 = 0 for 푗 ≥ 3.

iv. The entry is 푏푖,푗 = 1 for every 푖 = 푗 and for 푖 < 푗 the entry is 푏푖,푗 = 0.

By observing entries of 퐵4 for 푖 ≥ 푗, all entries of 퐵4 is listed in the following Table 3.2

Table 3.2: Elements of 퐵푛 Entries 퐵푛 The value of entries of 퐵푛

푏1,1 (1) 푙1,1 = 퐿(1,1)

푏2,2 (1) 푙2,2 = 퐿(2,2)

푏3,3 (1) 푙3,3 = 퐿(3,3)

푏4,4 (1) 푙4,4 = 퐿(4,4)

푏2,1 (1) 푙2,1 + (−푘) 푙2,2 = 퐿(2,1) − 푘퐿(2,2)

푏3,2 (1)푙3,2 + (−푘) 푙3,3 = 퐿(3,2) − 푘퐿(3,3)

푏4,3 (1)푙4,3 + (−푘) 푙4,4 = 퐿(4,3) − 푘퐿(4,4)

푏3,1 (1)푙3,1 + (−푘) 푙3,2 + (−1) 푙3,3 = 퐿(3,1) − 푘퐿(3,2) − 퐿(3,3)

푏4,2 (1)푙4,2 + (−푘) 푙4,3 + (−1) 푙4,4 = 퐿(4,2) − 푘퐿(4,3) − 퐿(4,4)

푏4,1 (1)푙4,1 + (−푘) 푙4,2 + (−1) 푙4,3 + (0) (1)푙4,4 = 퐿(4,1) − 푘퐿(4,2) − 퐿(4,3) ⋮ ⋮

푏푖,푗 퐿(푖, 푗) − 푘 퐿(푖, 푗 + 1) − 퐿(푖, 푗 + 2)

120 Irda Melina Zet et al. / IJMTT, 66(10), 116-122, 2020

So, Lah matrix and inverse of k-Fibonacci matrix multiplication gives a new matrix, which is 퐵푛 where the general definition is given by the following.

Definition 3.4. For every 푛 being natural number, the 푛 × 푛 matrix 퐵푛 with entry 퐵푛 = [푏푖,푗] for every 푖, 푗 = 1,2, … , 푛, is defined as follows

푏푖,푗 = 퐿(푖, 푗) − 푘 퐿(푖, 푗 + 1) − 퐿(푖, 푗 + 2).

It is easy to see that 푏푖,푗 = 1 for every 푖 = 푗, 푏푖,푗 = 0 for every 푖 < 푗. Moreover, for 푖 > 푗 then

푏푖,푗 = 퐿(푖, 푗) − 푘 퐿(푖, 푗 + 1) − 퐿(푖, 푗 + 2).

The following theorem is constructed from defining 퐵푛 matrix, Lah matrix, and k-Fibonacci matrix. Theorem 3.5 Lah matrix defined in Definition 2.2 can be stated as multiplication of k-Fibonacci 퐹푛(푘) in Definition 2.5 and 퐵푛 matrix in Definition 3.4 for 푛 and 푘 to be natural numbers such that 퐿푛 = 퐵푛 퐹푛(푘).

Proof. Since k-Fibonacci matrix 퐹푛(푘) has inverse it will be proven that −1 퐵푛 = 퐿푛 퐹푛 (푘). −1 Suppose 퐿푛 is Lah matrix, then the main diagonal of Lah matrix is 1. 퐹푛 (푘) is inverse of k-Fibonacci matrix, then the main diagonal of inverse of k-Fibonacci matrix is also 1. Multiplication of Lah matrix 퐿푛 and inverse of k-Fibonacci −1 matrix 퐹푛 (푘) resulting in a new matrix with main diagonal 1. Then, if 푖 = 푗 then 푏푖,푗 = 1 and if 푖 < 푗 then 푏푖,푗 = 0, and for 푖 > 2 then ′ ′ ′ ′ ′ 푏푖,푗 = 푙푖,푗 푓푗,푗(푘) + 푙푖,푗+1 푓푗+1,푗(푘) + 푙푖,푗+2 푓푗+2,푗(푘) + 푙푖,푗+3 푓푗+3,푗(푘) + ⋯ + 푙푖,푛 푓푛,푗(푘), 푛 ′ = ∑ 푙푖,푟 푓푟,푗(푘). 푟=1 −1 It can be concluded that 퐿푛 퐹푛 (푘) = 퐵푛. Hence, 퐿푛 = 퐵푛 퐹푛(푘). ∎

From Theorem 3.5, it can derived the following properties

Theorem 3.6. Let 퐿(푖, 푗) to be Lah numbers defined in Definition 2.1 and 퐹푘,푛 to be k-Fibonacci numbers defined in Definition 2.5 then for 푖 ≥ 푗 + 2 푖−2 2 퐿(푖, 푗) = 퐹푘,푖−푗+1 + (푖 − 푖 − 푘) 퐹푘,푖−푗 + ∑(퐿(푖, 푟) − 푘 퐿(푖, 푟 + 1) − 퐿(푖, 푟 + 2))퐹푘,푟−푗+1. 푟=푗 2 퐿(푖, 푗) = 퐹푘,푖−푗+1 + (푖 − 푖 − 푘) 퐹푘,푖−푗 for 푖 < 푗 + 2 and for 푖 < 푗 then 퐿(푖, 푗) = 0 for natural numbers 푖.

Proof. From Definition 3.4, 푏푖,푖 = 퐿(푖, 푖) = 1, 푏푖,푖−1 = 퐿(푖, 푖 − 1) − 푘퐿(푖, 푖) − 퐿(푖, 푖 + 1) = 푖(푖 − 1) − 푘(1) − 0 = 푖2 − 푖 − 푘, For 푗 ≤ 푟 ≤ 푖 − 2 then 푏푖,푡 = 퐿(푖, 푟) − 푘퐿(푖, 푟 + 1) − 퐿(푖, 푟 + 2).

Then, from Definition 2.5 and Theorem 3.5 퐿(푖, 푗) = 푙푖,푗 푖

= ∑ 푏푖,푟 푓푟,푗(푘) 푟=푗 푖

= ∑ 푏푖,푟 퐹푘,푟−푗+1 푟=푗 푖−2

= 푏푖,푖 퐹푘,푖−푗+1 + 푏푖,푖−1 퐹푘,푖−1−푗+1 + ∑ 푏푖,푟 퐹푘,푟−푗+1 푟=푗 푖−2 2 = 퐹푘,푖−푗+1 + (푖 − 푖 − 푘) 퐹푘,푖−푗 + ∑(퐿푖,푟 − 푘퐿푖,푟+1 − 퐿푖,푟+2) 퐹푘,푟−푗+1 푟=푗

121 Irda Melina Zet et al. / IJMTT, 66(10), 116-122, 2020

푖−2 2 퐿(푖, 푗) = 퐹푘,푖−푗+1 + (푖 − 푖 − 푘)퐹푘,푖−푗 + ∑(퐿(푖, 푟) − 푘퐿(푖, 푟 + 1) − 퐿(푖, 푟 + 2)) 퐹푘,푟−푗+1. ∎ 푟=푗

IV. CONCLUSION This article discusses about the relation of Lah matrix and k-Fibonacci matrix. From this relation, derived two kinds of Lah matrix factorization. The first factorization gives a new matrix that is obtained from multiplication of inverse of k-Fibonacci matrix and Lah matrix. The second factorization gives a new matrix that is yielded from multiplication of Lah matrix and inverse of k-Fibonacci matrix. These new matrices are difference. In addition, some properties that also states the relation of Lah numbers and k-Fibonacci numbers are also obtained.

REFERENCES [1] J. Engbers, D. Galvin, and C. Smyth, Restricted Stirling and Lah number matrices and their inverses, Journal of Combinatorial Theory Series A, 161 (2017), 1-26. [2] S. Falcon, The k-Fibonacci matrix and the Pascal matrix, Central European Journal of Mathematics, 9 (6), 2011, 1403–1410. [3] S. Falcon and A. Plaza, k-Fibonacci sequences modulo m, Chaos Solitons & Fractals, 41 (2009), 497-504. [4] S. Falcon and A. Plaza, On the Fibonacci k-numbers, Chaos Solitons & Fractals, 32 (2007), 1615-1624. [5] B. Guo and F. Qi, Six Proofs for an Identity of the Lah Numbers, Online Journal of Analytic Combinatorics, 10 (2015), 1-5. [6] V. E. Hoggat, Fibonacci and Lucas Numbers, Houghton–Mifflin, Palo Alto, CA., 1969. [7] T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley Interscience, New York, 2001. [8] G. Y. Lee and J. S. Kim, The linear algebra of the k-Fibonacci matrix, Linear Algebra and its Applications, Elsevier, 373 (2003), 75– 87. [9] I. Martinjak, Lah Number and Lindstrom’s Lemma, Comptes Rendus Mathematique, 356(2017), 1-4. [10] Mawaddaturrohmah and S. Gemawati, Relationship of Bell’s Polynomial Matrix and k-Fibonacci Matrix, American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS), 65(2020), 29-38. [11] R. Munir, Metode Numerik, Informatika : Bandung, 2008. [12] G. P. S. Rathore, A. A Wani, and K. Sisodiya, Matrix Representation of Generalized k-Fibonacci Sequence, OSR Journal of Mathematics, 12 (2016), 67–72. [13] T. Wahyuni, S. Gemawati, and Syamsudhuha, On some Identities of k-Fibonacci Sequences Modulo Ring 푍6 and 푍10, Applied Mathematical Sciences, 12 (2018), 441-448.

122