Curious harmony in asymmetric & nonlinear variant of Filbert and Lilbert matrices

Year 2024, , 724 - 734, 27.06.2024

Emrah Kılıç , Didem Ersanlı

https://doi.org/10.15672/hujms.1264983

Abstract

In this paper, we present a new analogue of the Filbert and Lilbert matrices whose indices have different asymmetric and nonlinear rules according to their row numbers. Explicit formulae are derived for the $LU$-decompositions, their inverses and the inverse of the main matrix as well as its determinant. To prove the claimed results we use backward induction method. The asymmetric variants of the Filbert and Lilbert matrices are obtained from our results for a particular $q$ value.

Keywords

Filbert matrix, Lilbert matrix, q-analogue, LU-decomposition, inverse matrix, determinant

References

• [1] Y. Amanbek, Z. Du, Y. Erlangga, C.M. da Fonseca, B. Kurmanbek and A. Pereira, Explicit determinantal formula for a class of banded matrices, Open Math. 18 (1), 1227-1229, 2020.
• [2] M. Andelic and C.M. da Fonseca, Some determinantal considerations for pentadiagonal matrices, Linear Multilinear Algebra 69 (16), 3121-3129, 2021.
• [3] J.E. Anderson and C. Berg, Quantum Hilbert matrices and orthogonal polynomials, J. Comput. Appl. Math. 223 (3), 723-729, 2009.
• [4] T. Arıkan, E. Kılıç and H. Prodinger, A nonsymmetrical matrix and its factorizations, Math. Slovaca 69 (4), 753-762, 2019.
• [5] L. Carlitz, The characteristic polynomial of a certain matrix of binomial coefficients, Fibonacci Q. 3, 81-89, 1965.
• [6] W. Chu and L. Claudio, Binomial determinant evaluations, Ann. Comb. 9 (4), 363- 377, 2005.
• [7] W. Chu, Finite differences and determinant identities, Linear Algebra Appl. 430, 215-228, 2009.
• [8] A. Edelman and G. Strang, Pascal matrices, Am. Math. Mon. 111, 189-197, 2004.
• [9] E. Kılıç and T. Arıkan, A class of non-symmetric band determinants with the Gaussian q-binomial coefficients, Quaest. Math. 40 (5), 645-660, 2017.
• [10] E. Kılıç and T. Arıkan, A nonlinear generalization of the Filbert matrix and its Lucas analogue, Linear Multilinear Algebra 67, 141-157, 2019.
• [11] E. Kılıç and D. Ersanlı, Harmony of asymmetric variants of the Filbert and Lilbert matrices in q-form, Math. Slovaca 73 (3), 633-642, 2023.
• [12] E. Kılıç, N. Ömür and S. Koparal, Nonlinear variants of the generalized Filbert and Lilbert matrices, Turk. J. Math. 44, 622-642, 2020.
• [13] E. Kılıç and H. Prodinger, Asymmetric generalizations of the Filbert matrix and variants, Publ. de l’Institut Math. (Beograd) (NS) 95 (109), 267-280, 2014.
• [14] M. Merca, A note on the determinant of a Toeplitz-Hessenberg matrix, Spec. Matrices 1, 10-16, 2013.
• [15] T. Richardson, The Filbert matrix, Fibonacci Q. 39 (3), 268-275, 2001.