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

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.