On the relations between antitrace, row/column sum of elements and generalized horadam sequence associated with a matrix

Sutaria, Kinnari A.; Desai, Narendrasinh B.

doi:10.47974/JDMSC-1910

Open Access Research Article

On the relations between antitrace, row/column sum of elements and generalized horadam sequence associated with a matrix

^*Kinnari A. SutariaCorresponding authorkinnari.patil@cvmu.edu.in kinnaripatil@adit.ac.inDepartment of MathematicsA. D. Patel Institute of TechnologyCharutar Vidya Mandal UniversityAnand, Gujarat, 388321, IndiaView full profile → , Narendrasinh B. Desainarendrasinh.desai@cvmu.edu.inDepartment of MathematicsA. D. Patel Institute of TechnologyCharutar Vidya Mandal UniversityAnand, Gujarat, 388321, IndiaView full profile →

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pp. 651–663Vol. 27Issue 2-BMarch 2024DOI: 10.47974/JDMSC-1910 Crossmark XML

Published Online:: 11 Apr 2024
Article type:: Research Article
Language:: EN
Article no.:: JDMSC-1910
Pages:: 651–663

Abstract

Using Cayley-Hamilton theorem for a given matrix A = [aij] ∈ M2(K), (K = ℝ or ℂ), we obtain the recurrence relations for Tra(An), suRi(An) and suCi(An); Tra(An+2) = Tr(A)Tra(An+1) – det(A) Tra(An) suRi(An+2) = Tr(A)suRi(An+1) – det(A)suRi(An) suCi(An+2) = Tr(A)suCi(An+1) – det(A) suCi(An) where Tra(A) = a12 + a21 is the sum of antidiagonal entries of A and suRi(A) or scCu(A) is the ith(i = 1, 2) row or column sum of A respectively. In doing so, the analogy between the above natural recurrences and the Horadam sequence is observed. Motivated by this, we define the Horadam sequence {Wn(a, b : Tr(A), det(A))} and Fibonacci sequence {Fn = Wn (0, 1; Tr(A), det(A))} associated with A. In this paper, Tra(An), {suRi(An)}, {suCi(An)}, {Wn} and {Fn} sequences are defined and their inter-relationships are extensively studied. We also obtain Binet’s formulae, generating functions, Cassini identities, Vajda’s identities and Catalan’s identities for these sequences. Further, identities are illustrated for particular values of trace and determinant of a matrix

Keywords

Antitrace of a matrix Cayley-Hamilton theorem Horadam sequence Fibonacci sequence Binet’s formula Generating function

Subject Classifications

Primary 11B3711B39 Secondary 15A15

References

[1] Durmus Bozkurt, Carlos M. Dafonseca, and FatIhyilmaz. The determinants of circulant and skew-circulant matrices with tribonacci numbers. Mathematical Sciences and Applications E Notes, : 67-75 (2014).
[2] Sergio Falcon and Angel Plaza. The ´ k-Fibonacci sequence and the Pascal 2-triangle. Chaos Solitons Fractals, : 38-49 (2007).
[3] Sergio Falcon and Angel Plaza. On the Fibonacci k-numbers. Chaos Solitons Fractals, : 1615-1624 (2007).
[4] Sergio Falcon and Angel Plaza. On the 3-dimensional k-Fibonacci spirals. Chaos Solitons Fractals, : 993-1003 (2008).
[5] Sergio Falcon and Angel Plaza. The metallic ratios as limits of complex valued transformations. Chaos Solitons Fractals, : 1-13 (2009).
[6] Sergio Falcon and Angel Plaza. On k-Fibonacci numbers of arithmetic indexes. Appl. Math. Comput., : 180-185 (2009).
[7] Sergio Falcon and Angel Plaza. On k-Fibonacci sequences and polynomials and their derivatives. Chaos Solitons Fractals, : 1005-1019 (2009).
[8] V. E. Hoggatt. Fibonacci and Lucas Numbers. The Fibonacci Association, California, USA (1969).
[9] F. Horadam. A generalized Fibonacci sequence. Amer. Math. Monthly, : 455-459 (1961).
[10] F. Horadam. Basic properties of a certain generalized sequence of numbers. Fibonacci Quart., : 161-176 (1965).
[11] F. Horadam. Generating functions for powers of a certain generalized sequence of numbers. Duke Math. J., :437-446 (1965).
[12] F. Horadam. Special properties of the sequence Wn(a, b; p, q). Fibonacci Quart., : 424-434 (1967).
[13] Hilkka Kankaanp¨a¨a and Jorma Kaarlo Merikoski. Two inequalities for the sum of elements of a matrix. Linear and Multilinear Algebra, :9-22 (1985).
[14] Dinesh J. Karia, Kailash M. Patil, and H. P. Singh. On the sum of powers of square matrices. Oper. Matrices, :221-229 (2019).
[15] Thomas Koshy. Fibonacci and Lucas numbers with applications. Pure and Applied Mathematics (New York). Wiley-Interscience, New York (2001).
[16] Mario Livio. The golden ratio. Broadway Books, New York. The story of phi, the world’s most astonishing number (2002).
[17] David London. Inequalities in quadratic forms. Duke Math. J., :511-522 (1966).
[18] David London. Two inequalities in nonnegative symmetric matrices. Pacific J. Math., : 515-536 (1966).
[19] Marvin Marcus and Morris Newman. The sum of the elements of the powers of a matrix. Pacific J. Math., : 627-635 (1962).
[20] Jorma Kaarlo Merikoski. On the trace and the sum of elements of a matrix. Linear Algebra Appl., : 177-185 (1984).
[21] Kailash M. Patil. On the trace of powers of square matrices. Hacet. J. Math. Stat., : 14-23 (2021).
[22] Angel Plaza and Sergio Falc´on. Combinatorial proofs of Honsberger-type identities. ´ Internat.J. Math. Ed. Sci. Tech., :785-792 (2008).
[23] Angel Plaza and Sergio Falc´on. Identities for generalized Fibonacci numbers: a combinatorial ´approach. Internat. J. Math. Ed. Sci. Tech., : 563-566 (2008).
[24] A. G. Shannon and A. F. Horadam. Generalized Fibonacci number triples. Amer. Math. Monthly, 80:187-190 (1973).
[25] Alexey Stakhov and Boris Rozin. On a new class of hyperbolic functions. Chaos Solitons Fractals, : 379-389 (2005).
[26] S. Vajda. Fibonacci & Lucas numbers and the golden section: Theory and its applications, E. Horwood Limited (1989).
[27] Kızılateş, Can & Kirlak, Selihan. A new generalization of Fibonacci and Lucas type sedenions, Journal of Discrete Mathematical Sciences and Cryptography, 1-12 (2022).
[28] Wanidchang, Kodchaphon & Kanasri, Narakorn Rompurk. On the period and the order of appearance of the sequence of Fibonacci polynomials modulo m, Journal of Discrete Mathematical Sciences and Cryptography, :181-192 (2022).

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