Chebyshev type lattice path weight polynomials by a constant term method

TL;DR

This paper introduces a constant term theorem for calculating weight polynomials of lattice paths with decorated weights, utilizing orthogonal polynomials and combinatorial methods, with applications in polymer physics.

Contribution

It presents a new constant term theorem for lattice path weight polynomials, an efficient method for explicit polynomial expressions of non-classical orthogonal polynomials, and a novel proof of Viennot's theorem using transfer matrix diagonalization.

Findings

Derived a constant term theorem for lattice path weights.

Developed a method for explicit orthogonal polynomial expressions.

Applied the method to polymer physics lattice path problems.

Abstract

We prove a constant term theorem which is useful for finding weight polynomials for Ballot/Motzkin paths in a strip with a fixed number of arbitrary `decorated' weights as well as an arbitrary `background' weight. Our CT theorem, like Viennot's lattice path theorem from which it is derived primarily by a change of variable lemma, is expressed in terms of orthogonal polynomials which in our applications of interest often turn out to be non-classical. Hence we also present an efficient method for finding explicit closed form polynomial expressions for these non-classical orthogonal polynomials. Our method for finding the closed form polynomial expressions relies on simple combinatorial manipulations of Viennot's diagrammatic representation for orthogonal polynomials. In the course of the paper we also provide a new proof of Viennot's original orthogonal polynomial lattice path theorem. The new proof is of interest because it uses diagonalization of the transfer matrix, but gets around difficulties that have arisen in past attempts to use this approach. In particular we show how to sum over a set of implicitly defined zeros of a given orthogonal polynomial, either by using properties of residues or by using partial fractions. We conclude by applying the method to two lattice path problems important in the study of polymer physics as models of steric stabilization and sensitized flocculation.