Hankel determinants and shifted periodic continued fractions (Q1631453)

For a sequence $A = (a_0,a_1,a_2,\ldots)$, the shifted (by $k$) Hankel matrices are defined as \[ \mathcal{H}_n^{(k)}(A) := (a_{i+j+k})_{0 \le i,j \le n-1}, \] while the shifted Hankel determinants are defined as \[ H_n^{(k)}(A) := \det\big(\mathcal{H}_n^{(k)}(A)\big), \] where $H_n^{(0)}(A)$ can be simply written as $H_n(A)$. There are many methods for evaluating Hankel determinants. When the generating function of the sequence $A$ satisfies a quadratic functional equation (which is the case for some path-counting numbers, such as the Calatan and Motzkin numbers), then continued fraction methods developed by \textit{I. M. Gessel} and \textit{G. Xin} [Electron. J. Comb. 13, No. 1, Research paper R53, 48 p. (2006; Zbl 1098.05006)] and \textit{R. A. Sulanke} and \textit{G. Xin} [Adv. Appl. Math. 40, No. 2, 149--167 (2008; Zbl 1141.05020)] may be used. This paper applies such methods to evaluate several shifted Hankel determinants (including those of the Catalan numbers), and to give alternative proofs of some former conjectures.