‘Classical’ convergence theorems for generalized continued fractions

Bruin, Marcel G. de

doi:10.1007/s11075-007-9115-1

Cited by 2 publications

(1 citation statement)

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“…In function theory, the so-called Jacobi-Perron algorithm and its variations are algorithms to expand a vector of power series in a continued fraction, where the partial numerators and denominators are vectors of polynomials. Algebraic and analytic aspects of vector continued fractions have been investigated in numerous works, such as [1,2,4,6,7,8,9,13,14,19,20,23,24,25,26]. Particularly important for us are the works of Kalyagin [13,14], in which he investigated spectral properties of banded Hessenberg operators, the problem of Hermite-Padé approximation to a vector of resolvent functions of the operator, and obtained the vector continued fraction for the vector of resolvent functions.…”

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confidence: 99%

Lattice paths, vector continued fractions, and resolvents of banded Hessenberg operators

López-García¹,

Prokhorov²

2022

Preprint

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We give a combinatorial interpretation of vector continued fractions obtained by applying the Jacobi-Perron algorithm to a vector of p ≥ 1 resolvent functions of a banded Hessenberg operator of order p + 1. The interpretation consists in the identification of the coefficients in the Laurent series expansion of the resolvent functions as weight polynomials associated with certain collections of lattice paths. In the scalar case p = 1 this reduces to the relation established by P. Flajolet between Jacobi continued fractions and Motzkin paths. A fundamental collection of lattice paths we consider is the collection of partial p-Łukasiewicz paths. We also discuss the particular case of Stieltjes-Rogers polynomials, also known in the literature as genetic sums, associated with partial p-Dyck paths and bi-diagonal Hessenberg operators.

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