IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 16, Issue 3 Ser. V (May – June 2020), PP 42-48 www.iosrjournals.org

Relation between Bernoulli Polynomial Matrix and k-Fibonacci Matrix

Hanif Handayani1, Syamsudhuha2, Sri Gemawati3 1,2,3(Department of mathematics, University of Riau, Indonesia) Corresponding Author: Hanif Handayani

Abstract: The Bernoulli polynomial matrix is expressed by Bn 푥 , with each entry being a Bernoulli polynomial. The k-Fibonacci matrix is represented by 퐹푛 푘 , with each entry being a k-Fibonacci number, whose first term is 0, the second term is 1, and the next term depends on a positive integer k. In this paper, we discuss about relation between the Bernoulli polynomial matrix and k-Fibonacci matrix. The results are define two new matrix, 퐶푛 푥 and 퐷푛 푥 such that Bn 푥 = 퐹푛 푘 퐶푛 푥 = 퐷푛 푥 퐹푛 푘 . Keyword: Bernoulli number, Bernoulli polynomial, Bernoulli matrix, Bernoulli polynomial matrix, k-Fibonacci number, k-Fibonacci matrix. ------Date of Submission: 20-06-2020 Date of Acceptance: 10-07-2020 ------

I. Introduction Bernoulli numbers are defined by Jacob Bernoulli (1654-1705) [1]. They first appeared in Ars Conjectandi, page 97, a famous treatise published in 1713. The Bernoulli polynomials are polynomials 퐵푛 of the degree 푛 which are convenient for integration by part as well as polynomials 푥푛 . They also are orthogonal to constant function. Bernoulli numbers and Bernoulli polynomials have been represented in matrices. The Bernoulli matrix is a lower trianguler matrix with each entry being a Bernoulli number, and if each entry being a Bernoulli polynomial, then that is Bernoulli polynomial matrix. Matrices and matrix theory are recently used in number theory and combinatorics. In particular, Bernoulli type lower triangular matrices are studied with Fibonacci, Pascal, and Stirling numbers, and other special numbers sequences. In 2008, Ernst [2] factorized (generalized) q-Bernoulli matrix by Pascal matrix and obtained some combinatorial identities. Can and Dagli [3] discussed generalized Bernoulli and Stirling matrices and obtained some of related combinatorial identities. Tuglu and Kus [4] studied q-Bernoulli and its properties. Earlier, in 2006, Zhang and Wang [5] disscussed relation between Bernoulli polynomial matrix and some of other matrices. They gave a product formula for the Bernoulli matrix and established several identities involving Fibonacci matrix, Stirling matrix, and Bernoulli polynomial matrix. The k-Fibonacci number was introduced by Falcon [6]. That is a number whose first term is 0, the second term is 1, and the next term depends on a positive integer k, and it can be represent in a matrix. The k-Fibonacci matrix is a lower trianguler matrix with each entry being a k-Fibonacci number. Wahyuni et al. [7] was introduced some identities of k-Fibonacci sequences modulo ring 푍6 and 푍10. Mawaddaturrohmah and S. Gemawati [8] have discussed about relationship of Bell’s polynomial matrix is expressed as 퐵푛 and k-Fibonacci matrix is expressed as 퐹푛 푘 , such that obtained two matrices, those are 푌푛 and 푍푛 , thus 퐵푛 = 퐹푛 푘 푌푛 = 푍푛 퐹푛 푘 , for any 푛, 푘 are natural number. In this paper, we discuss relation between Bernoulli polynomial matrix and k-Fibonacci matrix. Using two ideas of [2, 5] we define two matrices, those are 퐶푛 푥 and 퐷푛 푥 which states the relation between Bernoulli polynomial and k-Fibonacci matrices.

II. Preliminaries In this section, we introduce some definitions which are essential for the subsequent sections. We begin with some definitions and theories of Bernoulli and k-Fibonacci numbers and matrices. Firstly, we mention that Bernoulli numbers. Then using these numbers, a matrix can be delivered. This matrix is called Bernoulli matrix. Extending this matrix some matrices are obtained. Some definitions and theories related to Bernoulli and k-Fibonacci numbers and matrices that have been discussed by several authors [5, 9, 10, 11].

Definition 2.1. Bernoulli numbers are defined initial condition by 퐵0 = 1 and for any natural number 푛 hold

DOI: 10.9790/5728-1603054248 www.iosrjournals.org 42 | Page Relation between Bernoulli Polynomial Matrix and k-Fibonacci Matrix

푛−1 1 푛 + 1 퐵 = − 퐵 . 푛 푛 + 1 푘 푘 푘=0 The first few Bernoulli numbers are: 1 1 1 1 1 퐵 = 1, 퐵 = − , 퐵 = , 퐵 = 0, 퐵 = − , 퐵 = 0 , 퐵 = , 퐵 = 0, 퐵 = − . 0 1 2 2 6 3 4 30 5 6 42 7 8 30 For any 퐵푛 is a Bernoulli number, then 퐵푛 (푥) is a Bernoulli polynomial, that given in following definition.

Definition 2.2. Let 푛 be a natural number, the Bernoulli polynomials 퐵푛 (푥) are defined by 푛 푛 퐵 푥 = 퐵 푥푛−푘 . 푛 푘 푘 푘=0 The first few Bernoulli polynomials are: 퐵0 푥 = 1, 1 퐵 푥 = 푥 − , 1 2 1 퐵 푥 = 푥2 − 푥 + , 2 6 3 1 퐵 푥 = 푥3− 푥2 + 푥, 3 2 2 1 퐵 푥 = 푥4 − 2푥3+ 푥2 − , 4 30 5 5 1 퐵 푥 = 푥5 − 푥4 + 푥3 − 푥, 5 2 3 6 5 1 1 퐵 푥 = 푥6 − 3푥5 + 푥4 − 푥2 + . 6 2 2 42 Zhang [17] defined Bernoulli matrices by using Bernoulli numbers and polynomials. Also, they obtained factorization and some properties of Bernoulli matrices.

푡ℎ Definition 2.3. Let 퐵푛 be 푛 Bernoulli number and 퐵푛 (푥) be Bernoulli polynomial, (푛 + 1) × (푛 + 1) type Bernoulli matrix Bn = [푏푖,푗 ] and Bernoulli polynomial matrix Bn (푥) = [푏푖,푗 (푥)] defined respectively as follows 푖 퐵푖−푗 푖푓 푖 ≥ 푗, 푏푖,푗 = 푗 0 표푡ℎ푒푟푤푖푠푒, and 푖 퐵푖−푗 푥 푖푓 푖 ≥ 푗, 푏푖,푗 푥 = 푗 0 표푡ℎ푒푟푤푖푠푒.

Example 1. Bernoulli matrix B4 and Bernoulli polynomial matrix B4 푥 are 1 0 0 0 1 0 0 0 1 1 푥 − 1 0 0 − 1 0 0 2 2 B4 = and B4 푥 = 2 1 . 1 −1 1 푥 − 푥 + 2푥 − 1 1 0 6 0 6 1 3 2 1 3 − 3 3 2 1 3푥 − 3푥 + 3푥 − 0 2 2 1 푥 − 푥 + 푥 2 2 1 2 2 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, and 퐹푘,푛+1 = 푘 퐹푘,푛 + 퐹푘,푛−1 for 푛 ≥ 1. The first k-Fibonacci numbers are:

퐹푘,1 = 1,

퐹푘,2 = 푘,

DOI: 10.9790/5728-1603054248 www.iosrjournals.org 43 | Page Relation between Bernoulli Polynomial Matrix and k-Fibonacci Matrix

2 퐹푘,3 = 푘 + 1,

3 퐹푘,4 = 푘 + 2푘,

4 2 퐹푘,5 = 푘 + 3푘 + 1,

5 3 퐹푘,6 = 푘 + 4푘 + 3푘,

6 4 2 퐹푘,7 = 푘 + 5푘 + 6푘 + 1,

7 5 3 퐹푘,8 = 푘 + 6푘 + 10푘 + 4푘.

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 푓푖,푗 = 퐹푘,푖−푗 +1 if 푖 ≥ 푗, 0 otherwise. That is

1 0 0 0 0 0

퐹푘,2 1 0 0 0 0 퐹푘,3 퐹푘,2 1 0 0 0 Fn 푘 = . 퐹푘,4 퐹푘,3 퐹푘,2 1 0 0 ⋮ ⋮ ⋮ ⋮ ⋱ 0 퐹푘,푛 퐹푘,푛−1 퐹푘,푛−2 퐹푘,푛−3 ⋯ 1 Example 2. The 4 × 4 k-Fibonacci matrix is 1 0 0 0 푘 1 0 0 F (푘) = . 4 푘2 + 1 푘 1 0 푘3 + 2푘 푘2 + 1 푘 1 −1 Definition 2.6. Let F 푛(푘) be inverse matrix of the k-Fibonacci matrix, the 푛 × 푛 inverse k-Fibonacci −1 ′ matrix as lower 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, that is, 1 0 0 0 0 0 −푘 1 0 0 0 0

F −1(푘) = −1 −푘 1 0 0 0 . 푛 0 −1 −푘 1 0 0 ⋮ ⋱ ⋱ ⋱ ⋱ 0 0 0 0 −1 −푘 1 Example 3. The 4 × 4 inverse matrix of the k-Fibonacci matrix is 1 0 0 0 F −1(푘) = −푘 1 0 0 . 4 −1 −푘 1 0 0 −1 −푘 1

III. Relation Between Bernoulli Polynomial Matrix and k-Fibonacci Matrix In this section, we discuss relation between Bernoulli polynomial matrix and k-Fibonacci matrix by the similar way in [5]. We obtain definitions of new matrix 퐶푛 (푥) and 퐷푛 푥 . Furthermore, factorizations obtained on the Bernoulli polynomial matrix and k-Fibonacci matrix.

3.1 The First Factorization for Bernoulli Polynomial Matrix and k-Fibonacci Matrix We obtain relation between Bernoulli polynomial matrix and k-Fibonacci matrix by doing −1 multiplication between inverse of k-Fibonacci matrix 퐹푛 푘 and Bernoulli polynomial matrix Bn 푥 , for any 푛 −1 is natural number. The first step, for 푛 = 2, by multiply 퐹2 푘 and B2 푥 obtained 퐶2(푥), that is DOI: 10.9790/5728-1603054248 www.iosrjournals.org 44 | Page Relation between Bernoulli Polynomial Matrix and k-Fibonacci Matrix

1 0 1 0 1 0 −1 1 1 퐹2 푘 B2 푥 = 푥 − 1 = 푥 − 푘 − 1 = 퐶2(푥). −푘 1 2 2 −1 The second step, for 푛 = 3, by multiply 퐹3 (푘) and B3 푥 , we have 퐶3(푥) as follows 1 0 0 1 0 0 1 −1 푥 − 1 0 퐹3 푘 B3 푥 = −푘 1 0 2 1 −1 −푘 1 푥2 − 푥 + 2푥 − 1 1 6 1 0 0 1 푥 − 푘 − 1 0 = 2 1 5 푥2 − 푘 + 1 푥 + 푘 − 2푥 − 푘 − 1 1 2 6 = 퐶3(푥). The next step, for 푛 > 3, to simplify calculations of multiplication between inverse of k-Fibonacci matrix and Bernoulli polynomial matrix used Maple, such that obtained 퐶4 푥 , 퐶5 푥 , … , 퐶푛 (푥). Then, by look at each entry of 퐶4 푥 , we have that each entry of main diagonal is 1 (for 푖 = 푗 entry) and for 푖 > 푗 obtained as follows: st i. On the 1 row, then 푐1,1(푥) = 1, 푐1,푗 (푥) = 0 for 푗 ≥ 2. 1 ii. On the 2nd row, then 푐 (푥) = 푥 − 푘 − , 푐 (푥) = 1, 푐 (푥) = 0 for 푗 ≥ 3. 2,1 2 2,2 2,푗 1 5 iii. On the 3rd row, then 푐 (푥) = 푥2 − 푘 + 1 푥 + 푘 − , 푐 (푥) = 2푥 − 푘 − 1, 푐 (푥) = 1, 푐 (푥) = 3,1 2 6 3,2 3,3 3,푗 0 for 푗 ≥ 3. iv. Entry 푐푖,푗 (푥) = 1 for any 푖 = 푗 and 푐푖,푗 (푥) = 0 for 푖 < 푗. Thus, a list of all values of the 퐶4 푥 matrix entries are given in the following table.

Table 1: Element values for matrix 퐶푛 푥 Matrix 퐶푛 푥 Entry Matrix 퐶푛 푥 Entry Value

푐 (푥) 1 1,1 퐵 (푥) = 퐵 (푥) 1 0 0

푐 (푥) 2 2,2 퐵 푥 = 퐵 (푥) 2 0 0

푐 (푥) 3 3,3 퐵 푥 = 퐵 (푥) 3 0 0

푐 (푥) 4 4,4 퐵 푥 = 퐵 (푥) 4 0 0

푐 (푥) 2 1 2,1 2퐵 푥 − 푘퐵 푥 = 퐵 푥 + −푘 퐵 푥 1 0 1 1 1 0

푐 (푥) 3 2 3,2 3퐵 푥 − 푘퐵 푥 = 퐵 푥 + −푘 퐵 푥 1 0 2 1 2 0

푐 (푥) 4 3 4,3 4퐵 푥 − 푘퐵 푥 = 퐵 푥 + −푘 퐵 푥 1 0 3 1 3 0

푐 (푥) 3 2 1 3,1 3퐵 푥 − 2푘퐵 푥 − 퐵 푥 = 퐵 푥 + −푘 퐵 푥 + −1 퐵 푥 2 1 0 1 2 1 1 1 0

푐 (푥) 4 3 2 4,2 6퐵 푥 − 3푘퐵 푥 − 퐵 푥 = 퐵 푥 + −푘 퐵 푥 + −1 퐵 푥 2 1 0 2 2 2 1 2 0

푐 (푥) 4 3 2 4,1 4퐵 푥 − 3푘퐵 푥 − 2퐵 푥 = 퐵 푥 + −푘 퐵 푥 + −1 퐵 푥 3 2 1 1 3 1 2 1 1

⋮ ⋮

푐 (푥) 푖 푖 − 1 푖 − 2 푖,푗 퐵 푥 − 푘 퐵 푥 − 퐵 푥 푗 푖−푗 푗 푖−푗 −1 푗 푖−푗 −2

Furthermore, by look at each matrix entry and the matrix value of Table 1 obtained following definition. Definition 3.1. For every natural number 푛, it is defined an (푛 + 1) × (푛 + 1) matrix 퐶푛 푥 = [푐푖,푗 (푥)] with 푖, 푗 = 0,1,2, … , 푛 as follows DOI: 10.9790/5728-1603054248 www.iosrjournals.org 45 | Page Relation between Bernoulli Polynomial Matrix and k-Fibonacci Matrix

푖 푖 − 1 푖 − 2 푐 푥 = 퐵 푥 − 푘 퐵 푥 − 퐵 푥 . 푖,푗 푗 푖−푗 푗 푖−푗 −1 푗 푖−푗 −2 From Definition 3.1 obtained

푐0,0 푥 = 퐵0 푥 = 1,

푐0,푗 푥 = 0 for 푗 ≥ 1, 1 푐 푥 = 퐵 푥 − 푘퐵 푥 = 푥 − − 푘, 1,0 1 0 2 푐푖,0 푥 = 퐵푖 푥 − 푘퐵푖−1 푥 − 퐵푖−2 푥 for 푖 ≥ 2,

푐1,1 푥 = 퐵0 푥 = 1,

푐1,푗 푥 = 0 for 푗 ≥ 2.

From the definitions of Bernoulli polynomials matrix, k-Fibonacci matrix, and 퐶푛 푥 matrix, the following theorem can be derived.

Theorem 3.2. If Bn (푥) be a Bernoulli polynomial matrix, 퐹푛 푘 be a k-Fibonacci matrix, and 퐶푛 푥 be a new matrix in Definition 3.1, then Bn 푥 = 퐹푛 푘 퐶푛 푥 . Proof. Since 퐹푛 푘 be a k-Fibonacci matrix, then it is a invertible matrix. So that, we will be proven that

−1 퐶푛 푥 = 퐹푛 푘 Bn 푥 . −1 Let 퐹푛 푘 be a inverse of k-Fibonacci matrix and Bn 푥 be a Bernoulli polynomial matrix, then from Definition 2.6 and 2.3 we have the value of main diagonal of them is 1. The value of main diagonal 퐶푛 푥 matrix is 1. From Definition 3.1 obtained 푐푖,푗 (푥) = 1 for 푖 = 푗 and 푐푖,푗 (푥) = 0 for 푖 < 푗, thus for 푖 > 2, we get ′ 푐푖,푗 (푥) = 푓푖,푡 푘 푏푡,푗 푥 ′ ′ ′ ′ ′ = 푓푖,푖 푘 푏푖,푗 푥 + 푓푖,푖−1 푘 푏푖−1,푗 푥 + 푓푖,푖−2 푘 푏푖−2,푗 푥 + 푓푖,푖−3 푘 푏푖−3,푗 푥 + ⋯ + 푓푖,푛 푘 푏푛,푗 푥 , 푛 ′ = 푓푖,푡 푘 푏푡,푗 푥 . 푡=0

−1 Hence, 퐹푛 푘 Bn 푥 = 퐶푛 푥 such that Bn 푥 = 퐹푛 (푘)퐶푛 푥 .

3.2 The Second Factorization for Bernoulli Polynomial Matrix and k-Fibonacci Matrix We obtain relation between Bernoulli polynomial matrix and k-Fibonacci matrix by doing −1 multiplication between Bernoulli polynomial matrix Bn 푥 and inverse of k-Fibonacci matrix 퐹푛 푘 , for any −1 natural number 푛. The first step, for 푛 = 2, by multiply B2 푥 and 퐹2 푘 obtained 퐷2(푥), that is

1 0 1 0 1 0 −1 1 1 B2 푥 퐹2 푘 = 푥 − 1 = 푥 − 푘 − 1 = 퐷2(푥). 2 −푘 1 2

−1 The second step, for 푛 = 3, by multiply B3 푥 and 퐹3 (푘) , we have 퐷3(푥) as follows:

1 0 0 1 1 0 0 −1 푥 − 1 0 B3 푥 퐹3 푘 = 2 −푘 1 0 1 푥2 − 푥 + 2푥 − 1 1 −1 −푘 1 6 1 0 0 1 푥 − 푘 − 1 0 = 2 5 푥2 − 2푘 + 1 푥 + 푘 − 2푥 − 푘 − 1 1 6 = 퐷3(푥). The next step, for 푛 > 3, to simplify calculations of multiplication between Bernoulli polynomial matrix and inverse of k-Fibonacci matrix used Maple, such that obtained 퐷4 푥 , 퐷5 푥 , … , 퐷푛 (푥). Then, by look at each entry of 퐷4 푥 , we have that each entry of main diagonal is 1 (for 푖 = 푗 entry) and for 푖 > 푗 obtained as follows: st i. On the 1 row, then 푑1,1(푥) = 1, 푑1,푗 (푥) = 0 for 푗 ≥ 2. DOI: 10.9790/5728-1603054248 www.iosrjournals.org 46 | Page Relation between Bernoulli Polynomial Matrix and k-Fibonacci Matrix

1 ii. On the 2nd row, then 푑 푥 = 푥 − 푘 − , 푑 (푥) = 1, 푑 (푥) = 0 for 푗 ≥ 3. 2,1 2 2,2 2,푗 5 iii. On the 3rd row, then 푑 푥 = 푥2 − 2푘 + 1 푥 + 푘 − , 푑 푥 = 2푥 − 푘 − 1, 푑 푥 = 1, 3,1 6 3,2 3,3 푑3,푗 푥 = 0 for 푗 ≥ 3. iv. Entry 푑푖,푗 (푥) = 1 for any 푖 = 푗 and 푑푖,푗 (푥) = 0 for 푖 < 푗. Thus, a list of all values of the 퐷4 푥 matrix entries are given in the following table.

Table 2: Element values for matrix 퐷푛 푥 Matrix 퐷푛 푥 Entry Matrix 퐷푛 푥 Entry Value

푑 (푥) 1 1,1 퐵 (푥) = 퐵 (푥) 1 0 0

푑 (푥) 2 2,2 퐵 푥 = 퐵 (푥) 2 0 0

푑 (푥) 3 3,3 퐵 푥 = 퐵 (푥) 3 0 0

푑 (푥) 4 4,4 퐵 푥 = 퐵 (푥) 4 0 0

푑 (푥) 2 2 2,1 2퐵 푥 − 푘퐵 푥 = 퐵 푥 + −푘 퐵 푥 1 0 1 1 2 0

푑 (푥) 3 3 3,2 3퐵 푥 − 푘퐵 푥 = 퐵 푥 + −푘 퐵 푥 1 0 2 1 3 0

푑 (푥) 4 4 4,3 4퐵 푥 − 푘퐵 푥 = 퐵 푥 + −푘 퐵 푥 1 0 3 1 4 0

푑 (푥) 3 3 3 3,1 3퐵 푥 − 3푘퐵 푥 − 퐵 푥 = 퐵 푥 + −푘 퐵 푥 + −1 퐵 푥 2 1 0 1 2 2 1 3 0

푑 (푥) 4 4 4 4,2 6퐵 푥 − 4푘퐵 푥 − 퐵 푥 = 퐵 푥 + −푘 퐵 푥 + −1 퐵 푥 2 1 0 2 2 3 1 4 0

푑4,1(푥) 4퐵3 푥 − 6푘퐵2 푥 − 4퐵1 푥 4 4 4 = 퐵 푥 + −푘 퐵 푥 + −1 퐵 푥 1 3 2 2 3 1 4 + (0) 퐵 푥 4 0

⋮ ⋮

푑 (푥) 푖 푖 푖 푖,푗 퐵 푥 − 푘 퐵 푥 − 퐵 푥 푗 푖−푗 푗 + 1 푖−푗 −1 푗 + 2 푖−푗 −2

Furthermore, by look at each matrix entry and the matrix value of Table 2 obtained following definition. Definition 3.3. For every natural number 푛, it is defined an (푛 + 1) × (푛 + 1) matrix 퐷푛 푥 = [푑푖,푗 (푥)] with 푖, 푗 = 0,1,2, … , 푛 as follows 푖 푖 푖 푑 푥 = 퐵 푥 − 푘 퐵 푥 − 퐵 푥 . 푖,푗 푗 푖−푗 푗 + 1 푖−푗 −1 푗 + 2 푖−푗 −2 From Definition 3.3 obtained

푑0,0 푥 = 퐵0 푥 = 1,

푑0,푗 푥 = 0 for 푗 ≥ 1, 1 푑 푥 = 퐵 푥 − 푘퐵 푥 = 푥 − − 푘, 1,0 1 0 2 푖 푖 푑푖,0 푥 = 퐵푖 푥 − 푘 1 퐵푖−1 푥 − 2 퐵푖−2 푥 for 푖 ≥ 2,

푑1,1 푥 = 퐵0 푥 = 1,

DOI: 10.9790/5728-1603054248 www.iosrjournals.org 47 | Page Relation between Bernoulli Polynomial Matrix and k-Fibonacci Matrix

푑1,푗 푥 = 0 for 푗 ≥ 2.

From the definitions of Bernoulli polynomials matrix, k-Fibonacci matrix, and 퐷푛 푥 matrix, the following theorem can be derived.

Theorem 3.4. If Bn (푥) be a Bernoulli polynomial matrix, 퐹푛 푘 be a k-Fibonacci matrix, and 퐷푛 푥 be a new matrix in Definition 3.3, then Bn 푥 = 퐷푛 푥 퐹푛 푘 . Proof. Since 퐹푛 푘 be a k-Fibonacci matrix, then it is a invertible matrix. So that, we will be proven that −1 퐷푛 푥 =Bn 푥 퐹푛 푘 . −1 Let 퐹푛 푘 be a inverse of k-Fibonacci matrix and Bn 푥 be a Bernoulli polynomial matrix, then from Definition 2.6 and 2.3 we have the value of main diagonal of them is 1. The value of main diagonal 퐷푛 푥 matrix is 1. From Definition 3.3 obtained 푑푖,푗 (푥) = 1 for 푖 = 푗 and 푑푖,푗 (푥) = 0 for 푖 < 푗, thus for 푖 > 2, we get

′ 푑푖,푗 (푥) = 푏푖,푡 푥 푓푡,푗 푘 ′ ′ ′ ′ ′ = 푏푖,푗 푥 푓푗 ,푗 푘 + 푏푖,푗 +1 푥 푓푗 +1,푗 푘 + 푏푖,푗 +2 푥 푓푗 +2,푗 푘 + 푏푖,푗 +3 푥 푓푗+3,푗 푘 + ⋯ + 푏푖,푛 푥 푓푛,푗 푘 , 푛 ′ = 푏푖,푡 푥 푓푡,푗 푘 . 푡=0

−1 Hence, Bn 푥 퐹푛 푘 = 퐷푛 푥 such that Bn 푥 = 퐷푛 푥 퐹푛 푘 .

IV. Conclusion In this paper, the author discuss the relation between Bernoulli polynomial matrix and k- Fibonacci matrix, such that obtained two new matrices. Then, multiplication between the k-Fibonacci matrix and the first new matrix is the Bernoulli polynomial matrix. However, multiplication between the second new matrix and the k-Fibonacci matrix is the Bernoulli polynomial matrix too. Therefore, the first new matrix is not equal to the second new matrix and both of them are not commutative. For future research, it is necessary to think about the relation between q-Bernoulli polynomial matrix and k-Fibonacci matrix as well as the relation between Bernoulli polynomial and k-Fibonacci matrix.

References [1] P. Sebah dan X. Gourdon, Introduction on Bernoulli’s numbers, numbers.computation.free.fr/Constants/constants.html, 2002. [2] T. Ernst, q-Pascal and q-Bernoulli matrices and umbral approach, Department of Mathematics Uppsala Universty, 23, 2008. 1- 18. [3] M. Can and M. C. Dagli, Extended Bernoulli and Stirling matrices and related combinatorial identities, Linear Algebra and its Applications, 444, 2014, 114–131. [4] N. Tuglu dan S. Kus, q-Bernoulli matrices and their some properties, Gazi University Journal of Science, 28 (2), 2015, 269–273. [5] Z. Zhang dan J. Wang, Bernoulli matrix and its algebraic properties, Discrete Applied Mathematics, 154, 2006, 1622–1632. [6] S. Falcon dan A. Plaza, On the Fibonacci k-numbers, , Chaos Solitons & Fractals, 32, 2007, 1615-1624. [7] 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. [8] Mawaddaturrohmah dan 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. [9] S. Falcon, The k-Fibonacci matrix dan the Pascal matrix, Central European Journal of Mathematics, 9 (6), 2011, 1403–1410. [10] S. Falcon dan A. Plaza, k-Fibonacci sequences modulo m, Chaos Solitons & Fractals, 41, 2009, 497-504. [11] M. N. Lalin, Bernoulli Numbers, Junior Number Theory Seminar: Universty of Texas at Austin September 6th, 2005.

Hanif Handayani. "Relation between Bernoulli Polynomial Matrix and k-Fibonacci Matrix." IOSR Journal of Mathematics (IOSR-JM), 16(3), (2020): pp. 42-48.

DOI: 10.9790/5728-1603054248 www.iosrjournals.org 48 | Page