Hankel Determinants for a Class of Weighted Lattice Paths (2409.18609v1)

Abstract: 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\geq 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(dx^2 - bx - 1) + cx^2 + bx + 1}. ] 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 \geq 5), we utilize a class of shifted periodic continued fractions as defined by Wang-Xin-Zhai, thereby obtaining the results presented in this paper.