Hankel Determinants for a Class of Weighted Lattice Paths

In this paper, our primary goal is to calculate the Hankel determinants for a class of lattice paths, which are distinguished by the step set consisting of $\{(1,0),(2,0),(k-1,1),(-1,1)\}$, where the parameter $k\ge 4$. These paths are constrained to return to the $x$-axis and remain above the $x$-axis. When calculating for $k=4$, the problem essentially reduces to determining the Hankel determinant of $E(x)$, where $E(x)$ is defined as $$E(x)=\frac{a}{E(x)x^2(d{x}^2-bx-1)+c{x}^2+bx+1}\mathrm{.}$$ Our approach involves employing the Sulanke-Xin continued fraction transform to derive a set of recurrence relations, which in turn yield the desired results. For $k\ge 5$, we utilize a class of shifted periodic continued fractions as defined by Wang-Xin-Zhai, thereby obtaining the results presented in this paper.