Supernomial coefficients, polynomial identities and $q$-series

TL;DR

This paper introduces $q$-supernomial coefficients, explores their properties and combinatorial interpretations, and proves polynomial identities that unify and extend known boson-fermion identities in solvable lattice models.

Contribution

It proposes new $q$-analogues of coefficients, derives their properties, and establishes polynomial identities generalizing existing boson-fermion identities.

Findings

Derived recursion relations and symmetries of $q$-supernomial coefficients.

Provided a combinatorial interpretation via generalized Durfee dissection partitions.

Proved polynomial analogues of the Andrews-Gordon identities.

Abstract

$q$-Analogues of the coefficients of $x^a$ in the expansion of $\prod_{j=1}^N (1+x+...+x^j)^{L_j}$ are proposed. Useful properties, such as recursion relations, symmetries and limiting theorems of the ``$q$-supernomial coefficients'' are derived, and a combinatorial interpretation using generalized Durfee dissection partitions is given. Polynomial identities of boson-fermion-type, based on the continued fraction expansion of $p/k$ and involving the $q$-supernomial coefficients, are proven. These include polynomial analogues of the Andrews-Gordon identities. Our identities unify and extend many of the known boson-fermion identities for one-dimensional configuration sums of solvable lattice models, by introducing multiple finitization parameters.