Determinant Representations of Sequences: A Survey

• Autorzy: A. R. Moghaddamfar , S. Navid Salehy , S. Nima Salehy
• 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.

Słowa kluczowe

EN

determinant; generalized Pascal triangle; (Quasi) Toeplitz matrix; (Quasi) Pascal-like matrix; Fibonacci (Lucas; Jacobsthal and Pell) sequence

• Wydawca: De Gruyter Open
• Czasopismo: Special Matrices
• Rocznik: 2014
• Tom: 2
• Numer: 1
• Strony: 46-60

Daty

wydano

2014-01-01

otrzymano

2013-10-29

zaakceptowano

2014-04-03

online

2014-05-01

Twórcy

autor

A. R. Moghaddamfar

• Department of Mathematics, K. N. Toosi University of Technology, P. O. Box 16315-1618, Tehran, Iran

autor

S. Navid Salehy

• Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA

autor

S. Nima Salehy

• Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA

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.

Identyfikatory

DOI

10.2478/spma-2014-0005