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On Hessenberg and pentadiagonal determinants related with Fibonacci and Fibonacci-like numbers. (English) Zbl 1364.11041

Summary: In this paper, we establish several new connections between the generalizations of Fibonacci and Lucas sequences and Hessenberg determinants. We also give an interesting conjecture related to the determinant of an infinite pentadiagonal matrix with the classical Fibonacci and Gaussian Fibonacci numbers.

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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References:

[1] Anshelevich, M., Appell polynomials and their relatives, Int. Math. Res. Not., 65, 3469-3531 (2004) · Zbl 1086.33012
[2] Appell, P., Sur une classe de polynomes, Ann. Sci. l’E.N.S., 29, 119-144 (1880) · JFM 12.0342.02
[3] Bicknell-Johnson, M.; Spears, C., Classes of identities for the generalized Fibonacci numbers \(G_n = G_{n - 1} + G_{n - c}\) from matrices with constant valued determinants, Fibonacci Quart., 34, 121-128 (1996) · Zbl 0849.11020
[4] Bozkurt, D., On the spectral norms of the matrices connected to integer number sequences, Appl. Math. Comput., 219, 12, 6576-6579 (2013) · Zbl 1288.15024
[5] Cahill, N. D.; D’Errico, J. R.; Narayan, D. A.; Narayan, J. Y., Fibonacci determinants. The college math, J. Math. Assoc. Am., 33, 3, 221-225 (2002) · Zbl 1046.11007
[6] Cahill, N. D.; D’Errico, J. R.; Spence, J., Complex factorizations of the Fibonacci and Lucas numbers, Fibonacci Quart., 41, 1, 13-19 (2003) · Zbl 1056.11005
[7] Cahill, N. D.; Narayan, D. A., Fibonacci and Lucas numbers as tridiagonal matrix determinants, Fibonacci Quart., 42, 216-221 (2004) · Zbl 1080.11014
[8] Li, Ching, The maximum determinant of an \(n \times n\) lower Hessenberg \((0, 1)\) matrix, Linear Algebra Appl., 183, 147-153 (1993) · Zbl 0769.15008
[9] Civciv, H., A note on the determinant of five-diagonal matrices with Fibonacci numbers, Int. J. Contemp. Math. Sci., 3, 9, 419-424 (2008) · Zbl 1147.15004
[10] Esmaeili, M., More on the Fibonacci sequence and Hessenberg matrices, Integers, 6, 32 (2006) · Zbl 1123.11006
[11] Feng, J., Fibonacci identities via the determinant of tridiagonal matrix, Appl. Math. Comput., 217, 5978-5981 (2011) · Zbl 1218.11019
[12] Gauss, C. F., Theoria residuorum biquadraticorum, Commentatio secunda., Commun. Soc. Reg. Sci. Gottingen, 7, 1-34 (1832)
[13] Golub, G. H.; Van Loan, C. F., Matrix Computations (1996), Johns Hopkins University Press: Johns Hopkins University Press Baltimore, MD · Zbl 0865.65009
[14] Hogben, L., Handbook of Linear Algebra (2007), Chapman and Hall: Chapman and Hall Boca Raton · Zbl 1122.15001
[15] Horadam, A. F., Further appearence of the Fibonacci sequence, Fibonacci Quart., 1, 4, 41-42 (1963)
[16] İpek, A., On the spectral norms of circulant matrices with classical Fibonacci and Lucas numbers entries, Appl. Math. Comput., 217, 12, 6011-6012 (2011) · Zbl 1211.15028
[17] Koshy, T., Fibonacci and Lucas Numbers with Applications (2001), Wiley · Zbl 0984.11010
[18] Li, H. C., On Fibonacci-Hessenberg matrices and the Pell and Perrin numbers, Appl. Math. Comput., 218, 8353-8358 (2012) · Zbl 1251.15035
[19] Nalli, A.; Civciv, H., A generalization of tridiagonal matrix determinants, Fibonacci and Lucas numbers, Chaos Solitons Fract., 40, 1, 355-361 (2009) · Zbl 1197.15020
[20] Strang, G., Introduction to Linear Algebra (1998), Wellesley-Cambridge: Wellesley-Cambridge Wellesley, MA · Zbl 1067.15500
[21] Vein, R.; Dale, P., Determinants and Their Applications in Mathematical Physics. Determinants and Their Applications in Mathematical Physics, Appl. Math. Sci., vol. 134 (1998), Springer-Verlag: Springer-Verlag New York
[22] Yasar, M.; Bozkurt, D., Another proof of Pell identities by using the determinant of tridiagonal matrix, Appl. Math. Comput., 218, 6067-6071 (2012) · Zbl 1245.39002
[23] Wang, W.; Wang, T., Identities via Bell matrix and Fibonacci matrix, Discrete Appl. Math., 156, 14, 2793-2803 (2008) · Zbl 1152.15019
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