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Relation Between Bernoulli Polynomial Matrix and K-Fibonacci Matrix

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Relation Between Bernoulli Polynomial Matrix and K-Fibonacci Matrix 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).
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