Discrete Mathematics & Theoretical Computer Science |

- 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 $\gamma$-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 $a_k = ( 3 k+1) ! / (2k+1)!k!$.

Source: HAL:hal-01194684v1

Volume: DMTCS Proceedings vol. AI, Fifth Colloquium on Mathematics and Computer Science

Section: Proceedings

Published on: January 1, 2008

Imported on: May 10, 2017

Keywords: $\gamma$-operators,Hankel determinants,binomial coefficients,almost product form evaluations,differential equations,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]

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