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Supernomial coefficients, polynomial identities and \(q\)-series. (English) Zbl 0921.05007

Authors’ abstract: \(q\)-analogues of the coefficient of \(x^a\) in the expansion of \(\prod^N_{j=1} (1+x+x^2+ \cdots+ x^j)^{L_j}\) are proposed. Useful properties, such as recursion relations, symmetries and limiting theorems of the “\(q\)-supernomial coefficients” 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 given. 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 known lattice models by introducing multiple finitization parameters.

MSC:

05A30 \(q\)-calculus and related topics
05A19 Combinatorial identities, bijective combinatorics
11P84 Partition identities; identities of Rogers-Ramanujan type
33E99 Other special functions
82B23 Exactly solvable models; Bethe ansatz
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