Hankel Determinants for a Class of Weighted Lattice Paths

TL;DR

This paper computes Hankel determinants for a specific class of weighted lattice paths with complex step sets, using continued fraction transforms and recurrence relations, extending results for various parameter values.

Contribution

It introduces a novel method combining continued fractions and recurrence relations to evaluate Hankel determinants for weighted lattice paths with complex steps.

Findings

Derived explicit formulas for Hankel determinants for k=4.

Extended the method to k≥5 using shifted periodic continued fractions.

Provided recurrence relations for the determinants.

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.