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 $p\geq 1$ 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.