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HANKEL DETERMINANTS OF FACTORIAL FRACTIONS

Part of: Numerical linear algebra Basic linear algebra Polynomials and matrices

Published online by Cambridge University Press:  07 June 2021

WENCHANG CHU Open the ORCID record for WENCHANG CHU [Opens in a new window]
WENCHANG CHU*
Affiliation:
School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou (Henan), PR China and Current address: Department of Mathematics and Physics, University of Salento (P. O. Box 193), 73100 Lecce, Italy
*
e-mail: [email protected]
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Abstract

By making use of the Cauchy double alternant and the Laplace expansion formula, we establish two closed formulae for the determinants of factorial fractions that are then utilised to evaluate several determinants of binomial coefficients and Catalan numbers, including those obtained recently by Chammam [‘Generalized harmonic numbers, Jacobi numbers and a Hankel determinant evaluation’, Integral Transforms Spec. Funct. 30(7) (2019), 581–593].

Keywords

Hankel determinantbinomial determinantCauchy double alternantLaplace expansiondivided differencetelescoping methodpartial fraction decomposition

MSC classification

Primary: 15A15: Determinants, permanents, other special matrix functions
Secondary: 11C20: Matrices, determinants 65F40: Determinants
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 1 , February 2022 , pp. 46 - 57
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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References

Aigner, M., ‘Catalan-like numbers and determinants’,J. Combin. Theory Ser. A 87 (1999), 33–51.CrossRefGoogle Scholar
Aigner, M., ‘Catalan and other numbers: a recurrent theme’, in: Algebraic Combinatorics and Computer Science (eds. Crapo, H. and Senato, D.) (Springer, Milan, 2001), 347–390.CrossRefGoogle Scholar
Amdeberhan, T. and Zeilberger, D., ‘Determinants through the looking glass’,Adv. Appl. Math. 27 (2001), 225–230.CrossRefGoogle Scholar
Burchnall, J. L., ‘Some determinants with hypergeometric elements’, Q. J. Math. 3 (1952), 151–157.CrossRefGoogle Scholar
Chammam, W., ‘Generalized harmonic numbers, Jacobi numbers and a Hankel determinant evaluation’,Integral Transforms Spec. Funct. 30(7) (2019), 581–593.CrossRefGoogle Scholar
Chu, W., ‘Inversion techniques and combinatorial identities: a quick introduction to hypergeometric evaluations’,Math. Appl. 283 (1994), 31–57.Google Scholar
Chu, W., ‘Divided differences and symmetric functions’,Boll. Unione Mat. Ital. 2-B(3) (1999), 609–618.Google Scholar
Chu, W., ‘Binomial convolutions and determinant identities’,Discrete Math. 204(1–3) (1999), 129–153.CrossRefGoogle Scholar
Chu, W., ‘Generalizations of the Cauchy determinant’,Publ. Math. Debrecen 58(3) (2001), 353–365.Google Scholar
Chu, W., ‘The Cauchy double alternant and divided differences’,Electron. J. Linear Algebra 15 (2006), 14–21.CrossRefGoogle Scholar
Chu, W., ‘Finite differences and determinant identities’,Linear Algebra Appl. 430(1) (2009), 215–228.Google Scholar
Chu, W., ‘Telescopic generalizations for two 3 F 2-series identities’,Math. Slovaca 62(4) (2012), 689–694.CrossRefGoogle Scholar
Gould, H. W. and Hsu, L. C., ‘Some new inverse series relations’,Duke Math. J. 40 (1973), 885–891.CrossRefGoogle Scholar
Ostrowski, A. M., ‘On some determinants with combinatorial numbers,J. reine angew. Math. 216 (1964), 25–30.CrossRefGoogle Scholar
Tamm, U., ‘Some aspects of Hankel matrices in coding theory’,Electron. J. Combin. 8 (2001), Article ID #A1.CrossRefGoogle Scholar