Hankel Determinants for Some Common Lattice Paths

Details Hankel Determinants for Some Common Lattice Paths Hankel Determinants for Some Common Lattice Paths

2006-03-08

arxiv.org/abs/math/0603195v1

Mathematics

Combinatorics

14 pages, 2 figures, FPSAC 06

Scientific paper

For a single value of $\ell$, let $f(n,\ell)$ denote the number of lattice paths that use the steps $(1,1)$, $(1,-1)$, and $(\ell,0)$, that run from $(0,0)$ to $(n,0)$, and that never run below the horizontal axis. Equivalently, $f(n,\ell)$ satisfies the quadratic functional equation $F(x) = \sum_{n\ge 0}f(n,\ell) x^n = 1+x^{\ell}F(x)+x^2F(x)^2.$ Let $H_n$ denote the $n$ by $n$ Hankel matrix, defined so that $[H_n]_{i,j} = f(i+j-2,\ell)$. Here we investigate the values of such determinants where $\ell = 0,1,2,3$. For $\ell = 0,1,2$ we are able to employ the Gessel-Viennot-Lindstr\"om method. For the case $\ell=3$, the sequence of determinants forms a sequence of period 14, namely, $$ (\det(H_n))_{n \ge 1} = (1,1,0,0,-1,-1,-1,-1,-1,0,0,1,1,1,1,1,0,0,-1,-1,-1,...)$$ For this case we are able to use the continued fractions method recently introduced by Gessel and Xin. We also apply this technique to evaluate Hankel determinants for other generating functions satisfying a certain type of quadratic functional equation.

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