Orthogonal polynomials, Catalan numbers, and a general Hankel determinant evaluation (Q2427906)

Let \(A=(a_n)_{n \geq 0}\) be a sequence of complex numbers and \(\left(a_n(x)\right)_{n \geq 0}\) be a sequence of polynomials given by \(a_n(x) = \sum_{v=0}^n a_{n-v}x^v\), for \(n \geq 0\). The paper under review studies the Hankel determinant of the sequence of polynomials \(\left(a_{r+n}(x)\right)_{n \geq 0}\) where \(r \geq 0\) is a fixed integer. In Theorem 3, it is proved that Hankel determinant of these polynomials satisfy a three-term recurrence relation. In section 4, this result is illustrated with an example when \(A\) is the sequence of the Catalan numbers.