Ömer Eugeciouglu ; Timothy Redmond ; Charles Ryavec
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Evaluation of a Special Hankel Determinant of Binomial Coefficients
dmtcs:3569 -
Discrete Mathematics & Theoretical Computer Science,
January 1, 2008,
DMTCS Proceedings vol. AI, Fifth Colloquium on Mathematics and Computer Science
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https://doi.org/10.46298/dmtcs.3569
Evaluation of a Special Hankel Determinant of Binomial CoefficientsConference paper
Authors: Ömer Eugeciouglu 1; Timothy Redmond 2; Charles Ryavec 3
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Ömer Eugeciouglu;Timothy Redmond;Charles Ryavec
1 Department of Computer Science [Santa Barbara]
2 Stanford Medical Informatics
3 College of Creative Studies [Santa-Barbara]
This paper makes use of the recently introduced technique of γ-operators to evaluate the Hankel determinant with binomial coefficient entries ak=(3k)!/(2k)!k!. We actually evaluate the determinant of a class of polynomials ak(x) having this binomial coefficient as constant term. The evaluation in the polynomial case is as an almost product, i.e. as a sum of a small number of products. The γ-operator technique to find the explicit form of the almost product relies on differential-convolution equations and establishes a second order differential equation for the determinant. In addition to x=0, product form evaluations for x=35,34,32,3 are also presented. At x=1, we obtain another almost product evaluation for the Hankel determinant with ak=(3k+1)!/(2k+1)!k!.