Abstract
Two new asymmetric generalizations of the Filbert and Lilbert matrices constructed by the products of two Fibonacci and Lucas numbers are considered, with additional parameter settings. Explicit formulæ are derived for their LU-decompositions and inverses.
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J.E. Anderson, C. Berg, Quantum Hilbert matrices and orthogonal polynomials. J. Comput. Appl. Math. 223, 723–729 (2009)
A. Edelman, G. Strang, Pascal matrices. Am. Math. Mon. 111, 189–197 (2004)
L. Carlitz, The characteristic polynomial of a certain matrix of binomial coefficients. Fibonacci Q. 3, 81–89 (1965)
W. Chu, L. Di Claudio, Binomial determinant evaluations. Ann. Comb. 9, 363–377 (2005)
W. Chu, Finite differences and determinant identities. Linear Algebra Appl. 430, 215–228 (2009)
W. Chu, Binomial convolutions and determinant identities. Discrete Math. 204, 129–153 (1999)
W. Chu, X. Wang, W. Zhang, Partial fraction decomposition and determinant identities. Ars Comb. 109, 71–86 (2013)
E. Kılıç, H. Prodinger, A generalized Filbert matrix. Fibonacci Q. 48, 29–33 (2010)
E. Kılıç, H. Prodinger, The \(\mathit{q}\)-Pilbert matrix. Int. J. Comput. Math. 89, 1370–1377 (2012)
E. Kılıç, H. Prodinger, Variants of the Filbert matrix. Fibonacci Q. 51, 153–162 (2013)
E. Kılıç, H. Prodinger, Asymmetric generalizations of the Filbert matrix and variants. Publ. Inst. Math. (Beograd) (N.S) 95, 267–280 (2014)
E. Kılıç, H. Prodinger, The generalized \(q\)-Pilbert matrix. Math. Slovaca 64, 1083–1092 (2014)
E. Kılıç, H. Prodinger, The generalized Lilbert matrix. Period. Math. Hung. 73, 62–72 (2016)
E. Kılıç, H. Prodinger, The matrix of super Patalan numbers and its factorizations. Filomat 31, 2337–2344 (2017)
E. Kılıç, İ. Akkuş, G. Kızılaslan, A variant of the reciprocal super Catalan matrix with two additional parameters. Spec. Matrices 3, 163–168 (2015)
E. Kılıç, T. Arıkan, The generalized reciprocal super Catalan matrix. Turk. J. Math. 40, 960–972 (2016)
E. Kılıç, N. Ömür, S. Koparal, Y. Ulutaş, Two variants of the reciprocal super Catalan matrix. Kyungpook Math. J. 56, 409–418 (2016)
E. Kılıç, T. Arıkan, A nonlinear generalization of the Filbert matrix and its Lucas analogue. Accepted in Linear Multilinear Algebra, https://doi.org/10.1080/03081087.2017.1412393
M. Merca, A note on the determinant of a Toeplitz-Hessenberg matrix. Spec. Matrices 1, 10–16 (2013)
H. Prodinger, A generalization of a Filbert matrix with \(3\) additional parameters. Trans. R. Soc. S. Afr. 65, 169–172 (2010)
H. Prodinger, The reciprocal super Catalan matrix. Spec. Matrices 3, 111–117 (2015)
T. Richardson, The Filbert matrix. Fibonacci Q. 39, 268–275 (2001)
J. Zhou, J. Zhaolin, The spectral norms of \(g\)-circulant matrices with classical Fibonacci and Lucas numbers entries. Appl. Math. Comput. 233, 582–587 (2014)
J. Zhou, J. Zhaolin, Spectral norms of circulant-type matrices with Binomial coefficients and Harmonic numbers. Int. J. Comput. Methods 11, 1350076 (2014)
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The authors would like to thank the anonymous referee for carefully reading the paper.
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Kılıç, E., Koparal, S. & Ömür, N. New asymmetric generalizations of the Filbert and Lilbert matrices. Period Math Hung 78, 231–241 (2019). https://doi.org/10.1007/s10998-018-0253-0
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DOI: https://doi.org/10.1007/s10998-018-0253-0