Three determinant evaluations (Q2021617)

The authors evaluate by means of matrix decompositions three determinants whose elements are certain binomial sums. Among others they evaluate the determinant whose elements are \[ a_{ij}=\sum_{k=0}^{j} (-1)^{k} \binom{x+i}{k}\binom{y-i}{j-k}. \] As applications, the authors obtain evaluations of determinants whose elements involve quotients of rising factorials and Hermite, Laguerre and Legendre polynomials. This paper is inspired by the results found in [\textit{S. B. Ekhad} and \textit{D. Zeilberger}, in: Analytic number theory, modular forms and \(q\)-hypergeometric series. In honor of Krishna Alladi's 60th birthday, University of Florida, Gainesville, FL, USA, March 17--21, 2016. Cham: Springer. 253--265 (2017; Zbl 1391.11050)].