Hankel determinants of a linear combination of three successive Catalan numbers
The author evaluates the determinants of Hankel matrices, whose elements are a linear combination of three successive shifted Catalan numbers. Let \[ a_n=\alpha C_{n+k}+\beta C_{n+k+1}+\gamma C_{n+k+2},\;n\geq0,\;\alpha,\beta,\gamma \in \mathbb{C}, \] where \(C_n=\frac{\binom{2n}{n}}{n+1}\), \(n\geq0\), be Catalan numbers and \(k\) an arbitrary positive integer. By \(h_{n,k}(\gamma,\beta,\alpha)=\det(a_{i+j})_{0\leq i,j\leq n}\), he denotes the Hankel determinant of order \(n\) of \(a_0,a_1,\dots\). If \(\gamma\neq0\), the author shows that \(h_n(\gamma,\beta,\alpha)\) can be expressed in terms of some modified monic Jacobi polynomials. The particular case when \(\gamma=0\) and \(\beta\neq0\) is deduced, and, finally, some particular cases when \(\gamma\neq0\) are studied.
scientific article; zbMATH DE number 6164328