arXiv math/06031952006-03-08math.CO

几种常见格路的Hankel行列式

Hankel Determinants for Some Common Lattice Paths

Robert A. Sulanke, Guoce Xin

备注：: 14 pages, 2 figures, FPSAC 06

摘要：

对于$\ell$的单个值，令$f(n,\ell)$表示使用步骤$(1,1)$、$(1,-1)$和$(\ell,0)$、从$(0,0)$运行到$(n,0)$并且从不在水平轴下方运行的晶格路径的数量。等价地，$f(n,\ell)$满足二次函数方程$F(x) = \sum_{n\ge 0}f(n,\ell) x^n = 1+x^{\ell}F(x)+x^2F(x)^2.$令$H_n$表示$n$由$n$ Hankel矩阵，定义使得$[H_n]_{i,j} = f(i+j-2,\ell)$。在这里，我们调查的价值观，这样的决定因素，其中$\ell = 0,1,2,3$。对于$\ell = 0,1,2$，我们可以使用Gessel-Viennot-Lindström方法。对于$\ell=3$的情况，行列式序列形成周期为14的序列，即$$ (\det(H_n))_{n \ge 1} = (1,1,0,0,-1,-1,-1,-1,-1,0,0,1,1,1,1,1,0,0,-1,-1,-1,...)$$。对于这种情况，我们能够使用最近由Gessel和Xin引入的连分数方法。我们也应用这种技术来评估汉克尔行列式满足一定类型的二次函数方程的其他生成函数。

英文摘要：

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-Lindström 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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摘要：

英文摘要：

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