论文标题
晶格路径，矢量持续的分数以及带状的Hessenberg操作员的分解
Lattice paths, vector continued fractions, and resolvents of banded Hessenberg operators
论文作者
论文摘要
我们通过将jacobi-Perron算法应用于$ p \ geq 1 $ resolvent函数的矢量，对向量的持续分数进行了组合解释，该分数是$ p+p+1 $。该解释包括鉴定分解的功率序列扩展中的系数作为与上半平面中Lukasiewicz晶格路径相关的重量多项式。在标量案例中，$ p = 1 $这减少了P. flajolet和jacobi-stieltjes之间的G. Viennot建立的关系，继续分数，其功率系列扩展和Motzkin Paths。我们考虑三类的晶格路径，即上半平面中的Lukasiewicz路径，下半平面中的对称图像，以及第三类无限制的晶格路径，可以越过$ x $轴。我们通过相关的生成功率系列之间的关系建立了三个路径家族之间的关系。我们还讨论了由部分$ p $ -dyck路径形成的Lukasiewicz路径的子集合，其权重多项式在文献中被称为遗传和stieltjes-rogers多项式，并表达一定的双基因分子Hessenberg操作员。
We give a combinatorial interpretation of vector continued fractions obtained by applying the Jacobi-Perron algorithm to a vector of $p\geq 1$ resolvent functions of a banded Hessenberg operator of order $p+1$. The interpretation consists in the identification of the coefficients in the power series expansion of the resolvent functions as weight polynomials associated with Lukasiewicz lattice paths in the upper half-plane. In the scalar case $p=1$ this reduces to the relation established by P. Flajolet and G. Viennot between Jacobi-Stieltjes continued fractions, their power series expansion, and Motzkin paths. We consider three classes of lattice paths, namely the Lukasiewicz paths in the upper half-plane, their symmetric images in the lower half-plane, and a third class of unrestricted lattice paths which are allowed to cross the $x$-axis. We establish a relation between the three families of paths by means of a relation between the associated generating power series. We also discuss the subcollection of Lukasiewicz paths formed by the partial $p$-Dyck paths, whose weight polynomials are known in the literature as genetic sums or generalized Stieltjes-Rogers polynomials, and express certain moments of bi-diagonal Hessenberg operators.
