DOCSLIB.ORG

The Pascal Matrix Function and Its Applications to Bernoulli Numbers and Bernoulli Polynomials and Euler Numbers and Euler Polynomials

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Pascal Matrix Function and Its Applications to Bernoulli Numbers and Bernoulli Polynomials and Euler Numbers and Euler Polynomials The Pascal Matrix Function and Its Applications to Bernoulli Numbers and Bernoulli Polynomials and Euler Numbers and Euler Polynomials Tian-Xiao He ∗ Jeff H.-C. Liao y and Peter J.-S. Shiue z Dedicated to Professor L. C. Hsu on the occasion of his 95th birthday Abstract A Pascal matrix function is introduced by Call and Velleman in [3]. In this paper, we will use the function to give a unified approach in the study of Bernoulli numbers and Bernoulli poly- nomials. Many well-known and new properties of the Bernoulli numbers and polynomials can be established by using the Pascal matrix function. The approach is also applied to the study of Euler numbers and Euler polynomials. AMS Subject Classification: 05A15, 65B10, 33C45, 39A70, 41A80. Key Words and Phrases: Pascal matrix, Pascal matrix func- tion, Bernoulli number, Bernoulli polynomial, Euler number, Euler polynomial. ∗Department of Mathematics, Illinois Wesleyan University, Bloomington, Illinois 61702 yInstitute of Mathematics, Academia Sinica, Taipei, Taiwan zDepartment of Mathematical Sciences, University of Nevada, Las Vegas, Las Vegas, Nevada, 89154-4020. This work is partially supported by the Tzu (?) Tze foundation, Taipei, Taiwan, and the Institute of Mathematics, Academia Sinica, Taipei, Taiwan. The author would also thank to the Institute of Mathematics, Academia Sinica for the hospitality. 1 2 T. X. He, J. H.-C. Liao, and P. J.-S. Shiue 1 Introduction A large literature scatters widely in books and journals on Bernoulli numbers Bn, and Bernoulli polynomials Bn(x). They can be studied by means of the binomial expression connecting them, n X n B (x) = B xn−k; n ≥ 0: (1) n k k k=0 The study brings consistent attention of researchers working in combi- natorics, number theory, etc. In this paper, we will establish a matrix form of the above binomial expression using the following generalized Pascal matrix P [x] introduced by Call and Velleman in [3]: n P [x] := xn−k ; (2) k n;k≥0 which we also call the Pascal matrix function and P [1] is clearly the clas- sical Pascal matrix. Many well-known and new properties of Bernoulli numbers and Bernoulli polynomials can be obtained through this ma- T trix approach. In fact, by denoting B(x) = (B0(x);B1(x);:::) and T B = B(0) = (B1;B2;:::) , we may write (1) as B(x) = P [x]B(0): (3) P [x] is a homomorphic mapping from R to the infinite lower triangular matrices with real entries due to Theorem 2 shown [3]: Theorem 1.1 [3] (Homomorphism Theorem) For any x; y 2 R, P [x + y] = P [x]P [y]: (4) By noting ik ii − j = ; (5) k j j k − j one immediately find the (i; j) entry of the right-hand side of (4) can be written as i X i k xi−k yk−j; (6) k j k=j Pascal Matrix Function 3 while the (i; j) entry of the left-hand side of (4) is i−j i i i X i − j X ii − j (x+y)i−j = ykxi−j−k = xi−kyk−j; j j k j k − j k=0 k=j which is exactly (6) completing the proof of Theorem 1.1. In [4], iden- tity (5) is used to rederive several known properties and relationships involving the Bernoulli and Euler polynomials. In this paper, a uni- fied approach by means of P [x] is presented in the study of Bernoulli polynomials and numbers and Euler polynomials and numbers. For any square matrix A, the exponential of A is defined as the following matrix in a series form: 1 1 X 1 eA = I + A + A2 + A3 + ··· = Ak: 2! 3! k! k=0 We say the series converges for every A if each entry of eA converges. From [13] and the definition of eA, we have (α+β)A αA βA e = e e for any α; β 2 R; (eA)−1 = e−A; tA tA tA Dte = Ae = e A: (7) i i−j For n 2 N [ 0, denote Pn[x] = ( j x )0≤i;j≤n. [3] shows there exists xHn a unique Hn = (hi;j)0≤i;j≤n such that Pn[x] = e . In fact, from (7), one may have H = D exHn = D P [x] : (8) n x x=0 x n x=0 Denote H = (hi;j)0≤i;j. Then (4) of Theorem 1.1 can be proved in one line: P [x + y] = e(x+y)H = exH eyH = P [x]P [y]: It is easy to see that the entry hi;j of H and Hn, n = 0; 1;:::, are i if i = j + 1; h = (H) = ; (9) i;j i;j 0 otherwise: 4 T. X. He, J. H.-C. Liao, and P. J.-S. Shiue and the entry of Hk are i! if i = j + k; (Hk) = j! (10) i;j 0 otherwise: k Hence, Hn = 0 if k ≥ n. In next section, we will define new matrix functions from Pn[x], which and the matrix relationship (3) between the Bernoulli numbers and the Bernoulli polynomials through P [x] will be used to re-build some well-known properties of the Bernoulli numbers and the Bernoulli polynomials in Section 3. Finally, in Section 4, a similar approach is used to study the Euler numbers and Euler polynomials and their relationships with Bernoulli numbers and Bernoulli polynomials. 2 Matrix functions Ln[x] and L¯n[x] related to Pn[x] We now present the integrals of P [x]. From [1] (on Page 240), there R 1 holds expression of the linear mapping Ln : Pn[x] 7! 0 Pn[x]dx from matrix functions to matrices as n−1 Z 1 X Hk L = P [x]dx = n ; (11) n n (k + 1)! 0 k=0 xHn which is simply from the fact Pn[x] = e and the last formula in (7). Notice that L is a nonsingular lower triangular matrix with all main diagonal entries equal to 1. We now extend matrices (11) to matrix functions n−1 Z x Z x X Hk L [x] := P [t]dt = etHn dt = n xk+1; (12) n n (k + 1)! 0 0 k=0 R x which can be considered a mapping Ln[x]: Pn[t] 7! 0 Pn[t]dt associ- ated with any x 2 R. It is obvious Ln[1] ≡ Ln defined in (11). From (10) and noting k = i − j, the entries of Ln[x] can be given as 1 ixi−j+1; if i ≥ j ≥ 0; (L [x]) = i−j+1 j (13) n ij 0; otherwise: Pascal Matrix Function 5 Particularly, 1 i; if i ≥ j; (L ) = i−j+1 j (14) n ij 0; otherwise: We now give some properties of Ln[x] and its relationships with Pn[x]. Proposition 2.1 Let matrices Hn, Pn[x], and Ln[x] be defined as above. Then there holds HnLn[x] = Ln[x]Hn = Pn[x] − In: (15) Proof. By transferring the indexes, we have n X Hk H L [x] = L [x]H = n xk n n n n k! k=1 n X Hk = n xk − I = P [x] − I : k! n n n k=0 Inspired by [1], we define ¯ −1 Ln = Dn(−1)LnDn(−1) ; and ¯ −1 Ln[x] = Dn(−1)Ln[x]Dn(−1) ; (16) n−1 ¯ ¯ where Dn(−1) = diag(1; −1; 1; −1;:::; (−1) ). Hence, Ln = Ln[1]. −1 Note. Since Dn(−1)HnDn(−1) = −Hn, we have n−1 k −1 X Dn(−1)H Dn(−1) D (−1)P [x]D (−1) = n xk n n n k! k=0 n−1 k X (Dn(−1)HnDn(−1)) = xk k! k=0 n−1 k n−1 k X (−Hn) X (Hn) = xk = (−x)k = P [−x]: k! k! n k=0 k=0 ¯ The relationship between Ln[x] and Ln[x] can be shown below. 6 T. X. He, J. H.-C. Liao, and P. J.-S. Shiue ¯ Proposition 2.2 Let Ln[x] and Ln[x] be defined as before. Then there hold ¯ ¯ Ln[x] = −Ln[−x];Ln[x] = −Ln[−x]; and (17) ¯ ¯ Ln = −Ln[−1];Ln = −Ln[−1]: (18) Proof. From (16), Z x Z x Z −x ¯ −1 Ln[x] = Dn(−1) Pn[t]dt Dn(−1) = Pn[−t]dt = − Pn[t]dt; 0 0 0 which implies (17), and (18) follows as well. ¯ Using the above relationship between Ln[x] and Ln[x] and Proposi- tion 2.1, we immediately have ¯ Proposition 2.3 Let Hn, Pn[x], and Ln[x] be defined as before. Then n−1 k X (−Hn) L¯ [x] = xk+1; and (19) n (k + 1)! k=0 ¯ ¯ −HnLn[x] = −Ln[x]Hn = P [−x] − In; (20) n−1 k n−1 k X (−Hn) X H (−x)k+1 = − n xk+1: (21) (k + 1)! (k + 1)! k=0 k=0 Proof. From (16) and (17) ¯ −1 Ln[x] = Dn(−1)Ln[x]Dn(−1) n−1 k X (Dn(−1)HnDn(−1)) = xk+1 (k + 1)! k=0 n−1 k X (−Hn) = xk+1; (k + 1)! k=0 i.e., formula (19). Again, (17) and (15) yield Pascal Matrix Function 7 ¯ −HnLn[x] = HnLn[−x] = Pn[−x] − In and ¯ −Ln[x]Hn = Ln[−x]Hn = Pn[−x] − In: Substituting x = −x into (12) and noting (17) and (19), we have n−1 k n−1 k X (−Hn) X H xk+1 = L¯ [x] = −L [−x] = − n (−x)k+1; (k + 1)! n n (k + 1)! k=0 k=0 which implies (21).
Load more

Recommended publications
  • Chapter 4 Finite Difference Gradient Information
    Chapter 4 Finite Difference Gradient Information In the masters thesis Thoresson (2003), the feasibility of using gradient- based approximation methods for the optimisation of the spring and damper characteristics of an off-road vehicle, for both ride comfort and handling, was investigated. The Sequential Quadratic Programming (SQP) algorithm and the locally developed Dynamic-Q method were the two successive approximation methods used for the optimisation. The determination of the objective function value is performed using computationally expensive numerical simulations that exhibit severe inherent numerical noise. The use of forward finite differences and central finite differences for the determination of the gradients of the objective function within Dynamic-Q is also investigated. The results of this study, presented here, proved that the use of central finite differencing for gradient information improved the optimisation convergence history, and helped to reduce the difficulties associated with noise in the objective and constraint functions. This chapter presents the feasibility investigation of using gradient-based successive approximation methods to overcome the problems of poor gradient information due to severe numerical noise. The two approximation methods applied here are the locally developed Dynamic-Q optimisation algorithm of Snyman and Hay (2002) and the well known Sequential Quadratic 47 CHAPTER 4. FINITE DIFFERENCE GRADIENT INFORMATION 48 Programming (SQP) algorithm, the MATLAB implementation being used for this research (Mathworks 2000b). This chapter aims to provide the reader with more information regarding the Dynamic-Q successive approximation algorithm, that may be used as an alternative to the more established SQP method. Both algorithms are evaluated for effectiveness and efficiency in determining optimal spring and damper characteristics for both ride comfort and handling of a vehicle.
    [Show full text]
  • Stirling Matrix Via Pascal Matrix
    LinearAlgebraanditsApplications329(2001)49–59 www.elsevier.com/locate/laa StirlingmatrixviaPascalmatrix Gi-SangCheon a,∗,Jin-SooKim b aDepartmentofMathematics,DaejinUniversity,Pocheon487-711,SouthKorea bDepartmentofMathematics,SungkyunkwanUniversity,Suwon440-746,SouthKorea Received11May2000;accepted17November2000 SubmittedbyR.A.Brualdi Abstract ThePascal-typematricesobtainedfromtheStirlingnumbersofthefirstkinds(n,k)and ofthesecondkindS(n,k)arestudied,respectively.Itisshownthatthesematricescanbe factorizedbythePascalmatrices.AlsotheLDU-factorizationofaVandermondematrixof theformVn(x,x+1,...,x+n−1)foranyrealnumberxisobtained.Furthermore,some well-knowncombinatorialidentitiesareobtainedfromthematrixrepresentationoftheStirling numbers,andthesematricesaregeneralizedinoneortwovariables.©2001ElsevierScience Inc.Allrightsreserved. AMSclassification:05A19;05A10 Keywords:Pascalmatrix;Stirlingnumber;Stirlingmatrix 1.Introduction Forintegersnandkwithnk0,theStirlingnumbersofthefirstkinds(n,k) andofthesecondkindS(n,k)canbedefinedasthecoefficientsinthefollowing expansionofavariablex(see[3,pp.271–279]): n n−k k [x]n = (−1) s(n,k)x k=0 and ∗ Correspondingauthor. E-mailaddresses:[email protected](G.-S.Cheon),[email protected](J.-S.Kim). 0024-3795/01/$-seefrontmatter2001ElsevierScienceInc.Allrightsreserved. PII:S0024-3795(01)00234-8 50 G.-S. Cheon, J.-S. Kim / Linear Algebra and its Applications 329 (2001) 49–59 n n x = S(n,k)[x]k, (1.1) k=0 where x(x − 1) ···(x − n + 1) if n 1, [x] = (1.2) n 1ifn = 0. It is known that for an n, k 0, the s(n,k), S(n,k) and [n]k satisfy the following Pascal-type recurrence relations: s(n,k) = s(n − 1,k− 1) + (n − 1)s(n − 1,k), S(n,k) = S(n − 1,k− 1) + kS(n − 1,k), (1.3) [n]k =[n − 1]k + k[n − 1]k−1, where s(n,0) = s(0,k)= S(n,0) = S(0,k)=[0]k = 0ands(0, 0) = S(0, 0) = 1, and moreover the S(n,k) satisfies the following formula known as ‘vertical’ recur- rence relation: n− 1 n − 1 S(n,k) = S(l,k − 1).
    [Show full text]
  • The Original Euler's Calculus-Of-Variations Method: Key
    Submitted to EJP 1 Jozef Hanc, [email protected] The original Euler’s calculus-of-variations method: Key to Lagrangian mechanics for beginners Jozef Hanca) Technical University, Vysokoskolska 4, 042 00 Kosice, Slovakia Leonhard Euler's original version of the calculus of variations (1744) used elementary mathematics and was intuitive, geometric, and easily visualized. In 1755 Euler (1707-1783) abandoned his version and adopted instead the more rigorous and formal algebraic method of Lagrange. Lagrange’s elegant technique of variations not only bypassed the need for Euler’s intuitive use of a limit-taking process leading to the Euler-Lagrange equation but also eliminated Euler’s geometrical insight. More recently Euler's method has been resurrected, shown to be rigorous, and applied as one of the direct variational methods important in analysis and in computer solutions of physical processes. In our classrooms, however, the study of advanced mechanics is still dominated by Lagrange's analytic method, which students often apply uncritically using "variational recipes" because they have difficulty understanding it intuitively. The present paper describes an adaptation of Euler's method that restores intuition and geometric visualization. This adaptation can be used as an introductory variational treatment in almost all of undergraduate physics and is especially powerful in modern physics. Finally, we present Euler's method as a natural introduction to computer-executed numerical analysis of boundary value problems and the finite element method. I. INTRODUCTION In his pioneering 1744 work The method of finding plane curves that show some property of maximum and minimum,1 Leonhard Euler introduced a general mathematical procedure or method for the systematic investigation of variational problems.
    [Show full text]
  • Pascal Matrices Alan Edelman and Gilbert Strang Department of Mathematics, Massachusetts Institute of Technology [email protected] and [email protected]
    Pascal Matrices Alan Edelman and Gilbert Strang Department of Mathematics, Massachusetts Institute of Technology [email protected] and [email protected] Every polynomial of degree n has n roots; every continuous function on [0, 1] attains its maximum; every real symmetric matrix has a complete set of orthonormal eigenvectors. “General theorems” are a big part of the mathematics we know. We can hardly resist the urge to generalize further! Remove hypotheses, make the theorem tighter and more difficult, include more functions, move into Hilbert space,. It’s in our nature. The other extreme in mathematics might be called the “particular case”. One specific function or group or matrix becomes special. It obeys the general rules, like everyone else. At the same time it has some little twist that connects familiar objects in a neat way. This paper is about an extremely particular case. The familiar object is Pascal’s triangle. The little twist begins by putting that triangle of binomial coefficients into a matrix. Three different matrices—symmetric, lower triangular, and upper triangular—can hold Pascal’s triangle in a convenient way. Truncation produces n by n matrices Sn and Ln and Un—the pattern is visible for n = 4: 1 1 1 1 1 1 1 1 1 1 2 3 4 1 1 1 2 3 S4 = L4 = U4 = . 1 3 6 10 1 2 1 1 3 1 4 10 20 1 3 3 1 1 We mention first a very specific fact: The determinant of every Sn is 1. (If we emphasized det Ln = 1 and det Un = 1, you would write to the Editor.
    [Show full text]
  • Upwind Summation by Parts Finite Difference Methods for Large Scale
    Upwind summation by parts finite difference methods for large scale elastic wave simulations in 3D complex geometries ? Kenneth Durua,∗, Frederick Funga, Christopher Williamsa aMathematical Sciences Institute, Australian National University, Australia. Abstract High-order accurate summation-by-parts (SBP) finite difference (FD) methods constitute efficient numerical methods for simulating large-scale hyperbolic wave propagation problems. Traditional SBP FD operators that approximate first-order spatial derivatives with central-difference stencils often have spurious unresolved numerical wave-modes in their computed solutions. Recently derived high order accurate upwind SBP operators based non-central (upwind) FD stencils have the potential to suppress these poisonous spurious wave-modes on marginally resolved computational grids. In this paper, we demonstrate that not all high order upwind SBP FD operators are applicable. Numerical dispersion relation analysis shows that odd-order upwind SBP FD operators also support spurious unresolved high-frequencies on marginally resolved meshes. Meanwhile, even-order upwind SBP FD operators (of order 2; 4; 6) do not support spurious unresolved high frequency wave modes and also have better numerical dispersion properties. For all the upwind SBP FD operators we discretise the three space dimensional (3D) elastic wave equation on boundary-conforming curvilinear meshes. Using the energy method we prove that the semi-discrete ap- proximation is stable and energy-conserving. We derive a priori error estimate and prove the convergence of the numerical error. Numerical experiments for the 3D elastic wave equation in complex geometries corrobo- rate the theoretical analysis. Numerical simulations of the 3D elastic wave equation in heterogeneous media with complex non-planar free surface topography are given, including numerical simulations of community developed seismological benchmark problems.
    [Show full text]
  • Arxiv:1904.01037V3 [Math.GR]
    AN EFFECTIVE LIE–KOLCHIN THEOREM FOR QUASI-UNIPOTENT MATRICES THOMAS KOBERDA, FENG LUO, AND HONGBIN SUN Abstract. We establish an effective version of the classical Lie–Kolchin Theo- rem. Namely, let A, B P GLmpCq be quasi–unipotent matrices such that the Jordan Canonical Form of B consists of a single block, and suppose that for all k ě 0 the matrix ABk is also quasi–unipotent. Then A and B have a common eigenvector. In particular, xA, Byă GLmpCq is a solvable subgroup. We give applications of this result to the representation theory of mapping class groups of orientable surfaces. 1. Introduction Let V be a finite dimensional vector space over an algebraically closed field. In this paper, we study the structure of certain subgroups of GLpVq which contain “sufficiently many” elements of a relatively simple form. We are motivated by the representation theory of the mapping class group of a surface of hyperbolic type. If S be an orientable surface of genus 2 or more, the mapping class group ModpS q is the group of homotopy classes of orientation preserving homeomor- phisms of S . The group ModpS q is generated by certain mapping classes known as Dehn twists, which are defined for essential simple closed curves of S . Here, an essential simple closed curve is a free homotopy class of embedded copies of S 1 in S which is homotopically essential, in that the homotopy class represents a nontrivial conjugacy class in π1pS q which is not the homotopy class of a boundary component or a puncture of S .
    [Show full text]
  • High Order Gradient, Curl and Divergence Conforming Spaces, with an Application to NURBS-Based Isogeometric Analysis
    High order gradient, curl and divergence conforming spaces, with an application to compatible NURBS-based IsoGeometric Analysis R.R. Hiemstraa, R.H.M. Huijsmansa, M.I.Gerritsmab aDepartment of Marine Technology, Mekelweg 2, 2628CD Delft bDepartment of Aerospace Technology, Kluyverweg 2, 2629HT Delft Abstract Conservation laws, in for example, electromagnetism, solid and fluid mechanics, allow an exact discrete representation in terms of line, surface and volume integrals. We develop high order interpolants, from any basis that is a partition of unity, that satisfy these integral relations exactly, at cell level. The resulting gradient, curl and divergence conforming spaces have the propertythat the conservationlaws become completely independent of the basis functions. This means that the conservation laws are exactly satisfied even on curved meshes. As an example, we develop high ordergradient, curl and divergence conforming spaces from NURBS - non uniform rational B-splines - and thereby generalize the compatible spaces of B-splines developed in [1]. We give several examples of 2D Stokes flow calculations which result, amongst others, in a point wise divergence free velocity field. Keywords: Compatible numerical methods, Mixed methods, NURBS, IsoGeometric Analyis Be careful of the naive view that a physical law is a mathematical relation between previously defined quantities. The situation is, rather, that a certain mathematical structure represents a given physical structure. Burke [2] 1. Introduction In deriving mathematical models for physical theories, we frequently start with analysis on finite dimensional geometric objects, like a control volume and its bounding surfaces. We assign global, ’measurable’, quantities to these different geometric objects and set up balance statements.
    [Show full text]
  • Q-Pascal and Q-Bernoulli Matrices, an Umbral Approach Thomas Ernst
    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.
    [Show full text]
  • Numerical Stability & Numerical Smoothness of Ordinary Differential
    NUMERICAL STABILITY & NUMERICAL SMOOTHNESS OF ORDINARY DIFFERENTIAL EQUATIONS Kaitlin Reddinger A Thesis Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of MASTER OF ARTS August 2015 Committee: Tong Sun, Advisor So-Hsiang Chou Kimberly Rogers ii ABSTRACT Tong Sun, Advisor Although numerically stable algorithms can be traced back to the Babylonian period, it is be- lieved that the study of numerical methods for ordinary differential equations was not rigorously developed until the 1700s. Since then the field has expanded - first with Leonhard Euler’s method and then with the works of Augustin Cauchy, Carl Runge and Germund Dahlquist. Now applica- tions involving numerical methods can be found in a myriad of subjects. With several centuries worth of diligent study devoted to the crafting of well-conditioned problems, it is surprising that one issue in particular - numerical stability - continues to cloud the analysis and implementation of numerical approximation. According to professor Paul Glendinning from the University of Cambridge, “The stability of solutions of differential equations can be a very difficult property to pin down. Rigorous mathemat- ical definitions are often too prescriptive and it is not always clear which properties of solutions or equations are most important in the context of any particular problem. In practice, different definitions are used (or defined) according to the problem being considered. The effect of this confusion is that there are more than 57 varieties of stability to choose from” [10]. Although this quote is primarily in reference to nonlinear problems, it can most certainly be applied to the topic of numerical stability in general.
    [Show full text]
  • 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.
    [Show full text]
  • Factorizations for Q-Pascal Matrices of Two Variables
    Spec. Matrices 2015; 3:207–213 Research Article Open Access Thomas Ernst Factorizations for q-Pascal matrices of two variables DOI 10.1515/spma-2015-0020 Received July 15, 2015; accepted September 14, 2015 Abstract: In this second article on q-Pascal matrices, we show how the previous factorizations by the sum- mation matrices and the so-called q-unit matrices extend in a natural way to produce q-analogues of Pascal matrices of two variables by Z. Zhang and M. Liu as follows 0 ! 1n−1 i Φ x y xi−j yi+j n,q( , ) = @ j A . q i,j=0 We also find two different matrix products for 0 ! 1n−1 i j Ψ x y i j + xi−j yi+j n,q( , ; , ) = @ j A . q i,j=0 Keywords: q-Pascal matrix; q-unit matrix; q-matrix multiplication 1 Introduction Once upon a time, Brawer and Pirovino [2, p.15 (1), p. 16(3)] found factorizations of the Pascal matrix and its inverse by the summation and difference matrices. In another article [7] we treated q-Pascal matrices and the corresponding factorizations. It turns out that an analoguous reasoning can be used to find q-analogues of the two variable factorizations by Zhang and Liu [13]. The purpose of this paper is thus to continue the q-analysis- matrix theme from our earlier papers [3]-[4] and [6]. To this aim, we define two new kinds of q-Pascal matrices, the lower triangular Φn,q matrix and the Ψn,q, both of two variables. To be able to write down addition and subtraction formulas for the most important q-special functions, i.e.
    [Show full text]
  • Euler Matrices and Their Algebraic Properties Revisited
    Appl. Math. Inf. Sci. 14, No. 4, 1-14 (2020) 1 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.18576/amis/AMIS submission quintana ramirez urieles FINAL VERSION Euler Matrices and their Algebraic Properties Revisited Yamilet Quintana1,∗, William Ram´ırez 2 and Alejandro Urieles3 1 Department of Pure and Applied Mathematics, Postal Code: 89000, Caracas 1080 A, Simon Bol´ıvar University, Venezuela 2 Department of Natural and Exact Sciences, University of the Coast - CUC, Barranquilla, Colombia 3 Faculty of Basic Sciences - Mathematics Program, University of Atl´antico, Km 7, Port Colombia, Barranquilla, Colombia Received: 2 Sep. 2019, Revised: 23 Feb. 2020, Accepted: 13 Apr. 2020 Published online: 1 Jul. 2020 Abstract: This paper addresses the generalized Euler polynomial matrix E (α)(x) and the Euler matrix E . Taking into account some properties of Euler polynomials and numbers, we deduce product formulae for E (α)(x) and define the inverse matrix of E . We establish some explicit expressions for the Euler polynomial matrix E (x), which involves the generalized Pascal, Fibonacci and Lucas matrices, respectively. From these formulae, we get some new interesting identities involving Fibonacci and Lucas numbers. Also, we provide some factorizations of the Euler polynomial matrix in terms of Stirling matrices, as well as a connection between the shifted Euler matrices and Vandermonde matrices. Keywords: Euler polynomials, Euler matrix, generalized Euler matrix, generalized Pascal matrix, Fibonacci matrix,
    [Show full text]