Supernomial coefficients, polynomial identities and q-series

Abstract

q-Analogues of the coefficients of xa in the expansion ofΠj=1N(1 + x + ⋯ + xj)Lj 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.