On an identity of Gessel and Stanton and the new little Göllnitz identities

We show that an identity of Gessel and Stanton [I. Gessel, D. Stanton, Applications of q-Lagrange inversion to basic hypergeometric series, Trans. Amer. Math. Soc. 277 (1983) 197, Eq. (7.24)] can be viewed as a symmetric version of a recent analytic variation of the little Göllnitz identities. This is significant, since the Göllnitz-Gordon identities are considered the usual symmetric counterpart to little Göllnitz theorems. Is it possible, then, that the Gessel-Stanton identity is part of an infinite family of identities like those of Göllnitz-Gordon? Toward this end, we derive partners and generalizations of the Gessel-Stanton identity. We show that the new little Göllnitz identities enumerate partitions into distinct parts in which even-indexed (resp. odd-indexed) parts are even, and derive a refinement of the Gessel-Stanton identity that suggests a similar interpretation is possible. We study an associated system of q-difference equations to show that the Gessel-Stanton identity and its partner are actually two members of a three-element family.