[Paper Review] Lattice paths, q-multinomials and two variants of the Andrews-Gordon identities

Alexander Bérkovich, Peter Paule|ArXiv.org|Apr 4, 2001

Advanced Mathematical Identities9 references4 citations

TL;DR

This paper derives two new variants of the Andrews-Gordon identities using lattice path combinatorics and q-multinomial coefficients. By refining Bressoud's path model and leveraging finite polynomial identities from Foda, Quano, Kirillov, and Warnaar, the authors establish new multisum generating functions that generalize the original identities, with explicit formulas involving q-supernomial coefficients and modular constraints on partitions.

ABSTRACT

A few years ago Foda, Quano, Kirillov and Warnaar proposed and proved various finite analogs of the celebrated Andrews-Gordon identities. In this paper we use these polynomial identities along with the combinatorial techniques introduced in our recent paper to derive Garrett, Ismail, Stanton type formulas for two variants of the Andrews-Gordon identities.

Motivation & Objective

• To extend the Andrews-Gordon identities by deriving two new variants using combinatorial lattice path models and q-analogues.
• To generalize finite polynomial identities from Foda, Quano, Kirillov, and Warnaar into new multisum generating functions.
• To unify the structure of these identities through generalized q-multinomial coefficients (q-supernomial coefficients) with multiple finitization parameters.
• To establish a finite analog of the new variants using weighted lattice paths and peak height statistics.
• To provide a framework for a broader class of multisum identities parameterized by integer vectors (M₁,…,Mᵥ) via q-supernomial coefficients.

Proposed method

• Utilizes Bressoud’s lattice path model with NE, SE, and horizontal steps, defining path weights via peak positions and heights.
• Applies finite polynomial identities from Foda et al. to express generating functions as sums over lattice paths with fixed peak counts.
• Introduces generalized q-multinomials (q-supernomial coefficients) with ν finitization parameters to unify the structure of the identities.
• Employs recursive relations for polynomials $ C_{0,L}^{ u}(s,b,q) $ and $ G_{0,L}^{ u}(s,b,q) $ to derive finite versions of the identities.
• Uses the limit $ L \to \infty $ to recover infinite product forms, connecting to the original Andrews-Gordon identities via Jacobi triple product.
• Derives a unifying formula (5.7) for multisums with quadratic and linear terms in $ N_i $, using transformed parameters $ \tilde{\mathbf{M}} $ and q-supernomial coefficients.

Experimental results

Research questions

• RQ1 How can the Andrews-Gordon identities be generalized beyond the original partition-theoretic and analytic forms?
• RQ2 What is the combinatorial interpretation of new multisum generating functions with quadratic and linear terms in $ N_i $?
• RQ3 Can finite polynomial identities involving q-multinomials be used to derive new variants of the Andrews-Gordon identities?
• RQ4 What role do q-supernomial coefficients play in unifying different forms of these identities?
• RQ5 How do modular constraints on partition parts emerge from the lattice path and peak height statistics in the finite model?

Key findings

• The first variant of the Andrews-Gordon identity is derived as a limit of finite polynomial identities involving $ C_{0,L}^{ u}(s,b,q) $, with the generating function expressed as a sum over weighted lattice paths.
• The second variant is obtained by introducing a shift parameter $ M $, leading to a new generating function with a modified exponent $ N_1^2 + \cdots + N_\nu^2 - M(N_1 + \cdots + N_\nu) $, and a corresponding sum over $ s' $ with modular constraints.
• The key identity (4.9) expresses the generating function as a sum of rational functions over $ s' $, where each term corresponds to a product of a q-supernomial coefficient and an infinite product over partitions avoiding certain congruence classes modulo $ 2\nu+3 $.
• Setting $ M=1 $ in (4.9) yields identity (4.10), which matches (modulo a typo) identity (3.3) from [8], confirming consistency with prior work.
• The unifying formula (5.7) generalizes both variants into a single expression for multisums with arbitrary integer coefficients $ M_i $, using q-supernomial coefficients $ I_{s,b}^{\nu}(\mathbf{L},q) $.
• The finite analogs of the identities are established via recursive relations (1.13)–(1.15) and path counting with peak constraints, leading to the full generating function in the limit $ L \to \infty $.