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Evaluation of a Special Hankel Determinant of Binomial Coefficients

Abstract : This paper makes use of the recently introduced technique of $\gamma$-operators to evaluate the Hankel determinant with binomial coefficient entries $a_k = (3 k)! / (2k)! k!$. We actually evaluate the determinant of a class of polynomials $a_k(x)$ 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 $\gamma$-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 $x = \frac{3}{5}, \frac{3}{4}, \frac{3}{2}, 3$ are also presented. At $x=1$, we obtain another almost product evaluation for the Hankel determinant with $a_k = ( 3 k+1) ! / (2k+1)!k!$.
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Ömer Eugeciouglu, Timothy Redmond, Charles Ryavec. Evaluation of a Special Hankel Determinant of Binomial Coefficients. Fifth Colloquium on Mathematics and Computer Science, 2008, Kiel, Germany. pp.251-268. ⟨hal-01194684⟩

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