Lattice Paths, Vector Continued Fractions, and Resolvents of Banded Hessenberg Operators

IF 0.6 4区 数学 Q3 MATHEMATICS Computational Methods and Function Theory Pub Date : 2023-11-24 DOI:10.1007/s40315-023-00511-6
A. López-García, V. A. Prokhorov
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引用次数: 1

Abstract

We give a combinatorial interpretation of vector continued fractions obtained by applying the Jacobi–Perron algorithm to a vector of \(p\ge 1\) 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.

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点阵路径,向量连分数,和带状Hessenberg算子的解
将Jacobi-Perron算法应用于\(p+1\)阶带状Hessenberg算子的\(p\ge 1\)解函数向量,给出了向量连分式的组合解释。这种解释包括将解析函数的幂级数展开中的系数识别为与上半平面上的Lukasiewicz点阵路径相关的权多项式。在标量情况下\(p=1\)这可以归结为P. Flajolet和G. Viennot建立的Jacobi-Stieltjes连分式及其幂级数展开式和Motzkin路径之间的关系。我们考虑了三种晶格路径,即上半平面上的Lukasiewicz路径,它们在下半平面上的对称像,以及第三类允许穿过x轴的无限制晶格路径。通过相关的发电功率级数的关系,建立了三族路径之间的关系。我们还讨论了由偏p-Dyck路径构成的Lukasiewicz路径的子集合,其权多项式在文献中称为遗传和或广义stieltje - rogers多项式,并表示了双对角线Hessenberg算子的某些矩。
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Computational Methods and Function Theory
Computational Methods and Function Theory MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
3.20
自引率
0.00%
发文量
44
审稿时长
>12 weeks
期刊介绍: CMFT is an international mathematics journal which publishes carefully selected original research papers in complex analysis (in a broad sense), and on applications or computational methods related to complex analysis. Survey articles of high standard and current interest can be considered for publication as well.
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