The Generating Function of a Bi-Periodic Leonardo Sequence
DOI:
https://doi.org/10.52737/18291163-2024.16.07-1-8Keywords:
Leonardo Numbers, Generating Functions, Hessenberg Matrices, Recurrence Relations, DeterminantAbstract
In ``"A note on bi-periodic Leonardo sequence", the generating function for a certain bi-periodic Leonardo sequence is claimed. In this note, we show that the result is not correct. Based on the literature, we establish the correct identity. Possible periodic extensions for the Leonardo sequence are discussed, opening new avenues for results in the area.
References
M. Anđelić, Z. Du, C.M. da Fonseca and E. Kılıç, A matrix approach to some second-order difference equations with sign-alternating coefficients. J. Difference Equ. Appl., 26 (2020), no. 2, pp. 149-162. https://doi.org/10.1080/10236198.2019.1709180
M. Anđelić and C.M. da Fonseca, Determinantal representations for the number of subsequences without isolated odd terms. Notes Number Theory Discrete Math., 27 (2021), no. 4, pp. 116-121. https://doi.org/10.7546/nntdm.2021.27.4.116-121
M. Anđelić and C.M. da Fonseca, On the constant coefficients of a certain recurrence relation: A simple proof. Heliyon, 7 (2021), no. 8, E07764. https://doi.org/10.1016/j.heliyon.2021.e07764
M. Anđelić, C.M. da Fonseca and R. Mamede, On the number of P-vertices of some graphs. Linear Algebra Appl., 434 (2011), no. 2, pp. 514-525. https://doi.org/10.1016/j.laa.2010.09.017
M. Anđelić, C.M. da Fonseca and F. Yılmaz, The bi-periodic Horadam sequence and some perturbed tridiagonal 2-Toeplitz matrices: A unified approach. Heliyon, 8 (2022), no. 2, E08863. https://doi.org/10.1016/j.heliyon.2022.e08863
P.M.M. Catarino and E.V.P. Spreafico, A note on bi-periodic Leonardo sequence. Armen. J. Math., 16 (2024), no. 5, pp. 1-17. https://doi.org/10.52737/18291163-2024.16.5-1-17
J.F. Elliott, The characteristic roots of certain real symmetric matrices. Master's thesis, University of Tennessee, 1953.
C.M. da Fonseca, An identity between the determinant and the permanent of Hessenberg type-matrices. Czechoslovak Math. J., 61 (2011), no. 4, pp. 917-921. https://doi.org/10.1007/s10587-011-0059-1
C.M. da Fonseca and J. Petronilho, Explicit inverse of a tridiagonal k-Toeplitz matrix. Numer. Math., 100 (2005), no. 3, pp. 457-482. https://doi.org/10.1007/s00211-005-0596-3
C.M. da Fonseca and J. Petronilho, Explicit inverses of some tridiagonal matrices. Linear Algebra Appl., 325 (2001), no. 1-3, pp. 7-21.
S. Getu, Evaluating determinants via generating functions. Math. Mag., 64 (1991), pp. 45-53.
M.J.C. Gover, The eigenproblem of a tridiagonal 2-Toeplitz matrix. Linear Algebra Appl., 197/198 (1994), pp. 63-78. https://doi.org/10.1016/0024-3795(94)90481-2
A. Inselberg, On determinants of Toeplitz-Hessenberg matrices arising in power series. J. Math. Anal. Appl., 63 (1978), no. 2, pp. 347-353. https://doi.org/10.1016/0022-247x(78)90080-x
M. Jancić, Determinants and recurrence sequences. J. Integer Seq., 15 (2012), Article 12.3.5.
U. Leerawat and K. Daowsud, Determinants of some Hessenberg matrices with generating functions. Spec. Matrices, 11 (2023), no. 1, pp. 1-8. https://doi.org/10.1515/spma-2022-0170
D. Lehmer, Fibonacci and related sequences in periodic tridiagonal matrices. Fibonacci Quart., 13 (1975), 150-158.
F. Marcellán and J. Petronilho, Eigenproblems for tridiagonal 2-Toeplitz matrices and quadratic polynomial mappings. Linear Algebra Appl., 260 (1997), pp. 169-208. https://doi.org/10.1016/s0024-3795(97)80009-2
M. Merca, A note on the determinant of a Toeplitz-Hessenberg matrix. Spec. Matrices, 1 (2013), pp. 10-16. https://doi.org/10.2478/spma-2013-0003
P. Rózsa, On periodic continuants. Linear Algebra Appl., 2 (1969), no. 2, pp. 267-274. https://doi.org/10.1016/0024-3795(69)90030-5
D.E. Rutherford, Some continuant determinants arising in physics and chemistry. Proc. Roy. Soc. Edinburgh Sect. A, 62 (1947), no. 3, pp. 229-239. https://doi.org/10.1017/s0080454100006634
R. Vein and P. Dale, Determinants and their Applications in Mathematical Physics. Applied Mathematical Sciences, 134, New York, Springer, 1999.
L. Verde-Star, Polynomial sequences generated by infinite Hessenberg matrices. Spec. Matrices, 5 (2017), no. 1, pp. 64-72. https://doi.org/10.1515/spma-2017-0002
Downloads
Published
Issue
Section
License
Copyright (c) 2024 Armenian Journal of Mathematics
This work is licensed under a Creative Commons Attribution 4.0 International License.