Lattice paths, q-multinomials and two variants of the Andrews-Gordon identities

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. 1. Background and the first variant of the Andrews-Gordon identities In 1961, Gordon [12] found a natural generalization of the Rogers-Ramanujan partition theorem. Theorem 1. (Gordon) For all ν ≥ 1, 0 ≤ s ≤ ν, the partitions of N of the frequency form N = � j≥1 jfj with f1 ≤ s and fj + fj+1 ≤ ν, fj ≥ 0 (for all j ≥ 1) are equinumerous with the partitions of N into parts not congruent to 0 or ±(s + 1) modulo 2ν + 3. Thirteen years later, Andrews [1] proposed and proved the following analytic counterpart to Gordon’s theorem: Theorem 2. (Andrews) For all ν, s as in Theorem 1, and |q | < 1, 2