A note on the determinant of a Toeplitz-Hessenberg matrix
Mircea Merca
Special Matrices, Tome 1 (2013), p. 10-16 / Harvested from The Polish Digital Mathematics Library

The nth-order determinant of a Toeplitz-Hessenberg matrix is expressed as a sum over the integer partitions of n. Many combinatorial identities involving integer partitions and multinomial coefficients can be generated using this formula.

Publié le : 2013-01-01
EUDML-ID : urn:eudml:doc:267216
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     author = {Mircea Merca},
     title = {A note on the determinant of a Toeplitz-Hessenberg matrix},
     journal = {Special Matrices},
     volume = {1},
     year = {2013},
     pages = {10-16},
     zbl = {1291.15015},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_spma-2013-0003}
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Mircea Merca. A note on the determinant of a Toeplitz-Hessenberg matrix. Special Matrices, Tome 1 (2013) pp. 10-16. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_spma-2013-0003/

[1] G. E. Andrews, The Theory of Partitions, Addison-Wesley Publishing, 1976. | Zbl 0371.10001

[2] J. M. Bogoya, A. Böttcher, S. M. Grudsky, Asymptotics of Individual Eigenvalues of a Class of Large Hessenberg Toeplitz Matrices, Oper. Theory Adv. Appl. 220 (2012), 77–95. | Zbl 06069988

[3] J. M. Bogoya, A. Böttcher, S. M. Grudsky, E. A. Maksimenko, Eigenvectors of Hessenberg Toeplitz matrices and a problem by Dai, Geary, and Kadanoff, Linear Algebra Appl. 436 (2012), 3480–3492. [WoS]

[4] A. Z. Broder, The r-Stirling numbers, Discrete Math. 49 (1984), 241–259. [Crossref] | Zbl 0535.05006

[5] X. W. Chang, M. J. Gander, S. Karaa, Asymptotic properties of the QR factorization of banded Hessenberg–Toeplitz matrices. Numer. Linear Algebra Appl. 12 (2005), 659–682. | Zbl 1164.65332

[6] H. Chen, Bernoulli numbers via determinants, Internat. J. Math. Ed. Sci. Tech. 34 (2003), 291–297. [Crossref]

[7] K.-W. Chen, Inversion of Generating Functions using Determinants, J. Integer Seq. 10 (2007), Article 07.10.5. | Zbl 1174.11362

[8] R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics. Addison-Wesley, New York, 1994.

[9] A. Inselberg, On determinants of Toeplitz-Hessenberg matrices arising in power series, J. Math. Anal. Appl. 63 (1978), 347–353. [Crossref] | Zbl 0379.15003

[10] A. J. E. M. Janssen, Asymptotics of the Perron-Frobenius eigenvalue of nonnegative Hessenberg-Toeplitz matrices, IEEE Trans. Inf. Theor., 35 (1989), 1340–1344. | Zbl 0696.94007

[11] I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Clarendon Press, Oxford, 1995. | Zbl 0824.05059

[12] P. A. MacMahon, Combinatory analysis. Two volumes (bound as one), Chelsea Publishing Co., New York, 1960.

[13] P. Mongelli, Combinatorial interpretations of particular evaluations of complete and elementary symmetric functions, Electron. J. Combin. 19 (2012), paper 12, 23 pp. | Zbl 1243.05040

[14] T. Muir, The theory of determinants in the historical order of development. Four volumes, Dover Publications, New York, 1960.

[15] R. Van Malderen, Non-recursive expressions for even-index Bernoulli numbers: A remarkable sequence of determinants, arXiv:math/0505437v1, 2005.