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Relation Between Bernoulli Polynomial Matrix and K-Fibonacci Matrix

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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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    U.U.D.M. Report 2008:23 q-Pascal and q-Bernoulli matrices, an umbral approach Thomas Ernst Department of Mathematics Uppsala University q- PASCAL AND q-BERNOULLI MATRICES, AN UMBRAL APPROACH THOMAS ERNST Abstract A q-analogue Hn,q ∈ Mat(n)(C(q)) of the Polya-Vein ma- trix is used to define the q-Pascal matrix. The Nalli–Ward–AlSalam (NWA) q-shift operator acting on polynomials is a commutative semi- group. The q-Cauchy-Vandermonde matrix generalizing Aceto-Trigiante is defined by the NWA q-shift operator. A new formula for a q-Cauchy- Vandermonde determinant with matrix elements equal to q-Ward num- bers is found. The matrix form of the q-derivatives of the q-Bernoulli polynomials can be expressed in terms of the Hn,q. With the help of a new q-matrix multiplication certain special q-analogues of Aceto- Trigiante and Brawer-Pirovino are found. The q-Cauchy-Vandermonde matrix can be expressed in terms of the q-Bernoulli matrix. With the help of the Jackson-Hahn-Cigler (JHC) q-Bernoulli polynomials, the q-analogue of the Bernoulli complementary argument theorem is ob- tained. Analogous results for q-Euler polynomials are obtained. The q-Pascal matrix is factorized by the summation matrices and the so- called q-unit matrices. 1. Introduction In this paper we are going to find q-analogues of matrix formulas from two pairs of authors: L. Aceto, & D. Trigiante [1], [2] and R. Brawer & M.Pirovino [5]. The umbral method of the author [11] is used to find natural q-analogues of the Pascal- and the Cauchy-Vandermonde matrices.
  • The Images of Non-Commutative Polynomials Evaluated on 2 × 2 Matrices

    The Images of Non-Commutative Polynomials Evaluated on 2 × 2 Matrices

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 140, Number 2, February 2012, Pages 465–478 S 0002-9939(2011)10963-8 Article electronically published on June 16, 2011 THE IMAGES OF NON-COMMUTATIVE POLYNOMIALS EVALUATED ON 2 × 2 MATRICES ALEXEY KANEL-BELOV, SERGEY MALEV, AND LOUIS ROWEN (Communicated by Birge Huisgen-Zimmermann) Abstract. Let p be a multilinear polynomial in several non-commuting vari- ables with coefficients in a quadratically closed field K of any characteristic. It has been conjectured that for any n, the image of p evaluated on the set Mn(K)ofn by n matrices is either zero, or the set of scalar matrices, or the set sln(K) of matrices of trace 0, or all of Mn(K). We prove the conjecture for n = 2, and show that although the analogous assertion fails for completely homogeneous polynomials, one can salvage the conjecture in this case by in- cluding the set of all non-nilpotent matrices of trace zero and also permitting dense subsets of Mn(K). 1. Introduction Images of polynomials evaluated on algebras play an important role in non- commutative algebra. In particular, various important problems related to the theory of polynomial identities have been settled after the construction of central polynomials by Formanek [F1] and Razmyslov [Ra1]. The parallel topic in group theory (the images of words in groups) also has been studied extensively, particularly in recent years. Investigation of the image sets of words in pro-p-groups is related to the investigation of Lie polynomials and helped Zelmanov [Ze] to prove that the free pro-p-group cannot be embedded in the algebra of n × n matrices when p n.
  • Relation Between Lah Matrix and K-Fibonacci Matrix

    Relation Between Lah Matrix and K-Fibonacci Matrix

    International Journal of Mathematics Trends and Technology Volume 66 Issue 10, 116-122, October 2020 ISSN: 2231 – 5373 /doi:10.14445/22315373/IJMTT-V66I10P513 © 2020 Seventh Sense Research Group® 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.