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

Abstract

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.