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2014 | 2 | 1 | 46-60
Tytuł artykułu

Determinant Representations of Sequences: A Survey

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
This is a survey of recent results concerning (integer) matrices whose leading principal minors are well-known sequences such as Fibonacci, Lucas, Jacobsthal and Pell (sub)sequences. There are different ways for constructing such matrices. Some of these matrices are constructed by homogeneous or nonhomogeneous recurrence relations, and others are constructed by convolution of two sequences. In this article, we will illustrate the idea of these methods by constructing some integer matrices of this type.
Wydawca
Czasopismo
Rocznik
Tom
2
Numer
1
Strony
46-60
Opis fizyczny
Daty
wydano
2014-01-01
otrzymano
2013-10-29
zaakceptowano
2014-04-03
online
2014-05-01
Twórcy
  • Department of Mathematics, K. N. Toosi University of Technology, P. O. Box 16315-1618, Tehran, Iran, moghadam@kntu.ac.ir
Bibliografia
  • [1] R. Bacher, Determinants of matrices related to the Pascal triangle, J. Théor. Nombres Bordeaux, 14(1)(2002), 19-41.
  • [2] P. F. Byrd, Problem B-12: A Lucas determinant, Fibonacci Quart., 1(4)(1963), 78.
  • [3] N. D. Cahill, J. R. D’Errico, D. A. Narayan and J. Y. Narayan, Fibonacci determinants, College Math. J., 33(3)(2002), 221-225.[Crossref]
  • [4] N. D. Cahill, J. R. D’Errico and J. P. Spence, Complex factorizations of the Fibonacci and Lucas numbers, Fibonacci Quart., 41(1)(2003), 13-19.
  • [5] N. D. Cahill and D. A. Narayan, Fibonacci and Lucas numbers as tridiagonal matrix determinants, Fibonacci Quart., 42(3)(2004), 216-221.
  • [6] G. S. Cheon, S. G. Hwang, S. H. Rim and S. Z. Song, Matrices determined by a linear recurrence relation among entries, Special issue on the Combinatorial Matrix Theory Conference (Pohang, 2002), Linear Algebra Appl. 373 (2003), 89-99.
  • [7] K. Griffin, J. L. Stuart and M. J. Tsatsomeros, Noncirculant Toeplitz matrices all of whose powers are Toeplitz, CzechoslovakMath. J., 58(133)(4)(2008), 1185-1193.
  • [8] A. R. Moghaddamfar, K. Moghaddamfar and H. Tajbakhsh, New families of integer matrices whose leading principal minors form some well-known sequences, Electron. J. Linear Algebra, 22(2011), 598-619.
  • [9] A. R. Moghaddamfar and S. M. H. Pooya, Generalized Pascal triangles and Toeplitz matrices, Electron. J. Linear Algebra, 18(2009), 564-588.
  • [10] A. R. Moghaddamfar, S. M. H. Pooya, S. Navid Salehy and S. Nima Salehy, Fibonacci and Lucas sequences as the principal minors of some infinite matrices, J. Algebra Appl., 8(6)(2009), 869-883.[WoS][Crossref]
  • [11] A. R. Moghaddamfar, S. Rahbariyan, S. Navid Salehy and S. Nima Salehy, Some infinite matrices whose leading principal minors are well-known sequences, Util. Math., (to appear)
  • [12] A. R. Moghaddamfar and H. Tajbakhsh, Lucas numbers and determinants, Integers, 12(1)(2012), 21-51.
  • [13] A. R. Moghaddamfar and H. Tajbakhsh, More determinant representations for sequences, J. Integer Seq., 17(5)(2014), Article 14.5.6, 16 pp.
  • [14] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences. Published electronically at http://oeis.org, 2013.
  • [15] G. Strang, Introduction to Linear Algebra, Third Edition. Wellesley-Cambridge Press, 1993.
  • [16] G. Strang and K. Borre, Linear Algebra, Geodesy, and GPS, Wellesley-Cambridge Press, 1997.
  • [17] M. Tan, Matrices associated to biindexed linear recurrence relations, Ars Combin., 86 (2008), 305-319.
  • [18] S. Vajda, Fibonacci and Lucas numbers, and the golden section: Theory and applications, With chapter XII by B. W. Conolly. Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, 1989. 190 pp.
  • [19] Y. Yang and M. Leonard, Evaluating determinants of convolution-like matrices via generating functions, Int. J. Inf. Syst. Sci., 3(4)(2007), 569-580.
  • [20] H. Zakrajšek and M. Petkovšek, Pascal-like determinants are recursive, Adv. in App Math., 33(3)(2004), 431-450.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_spma-2014-0005
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