Advances in Applied Mathematics
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.
Savage, Carla D., Andrew Sills.
"On an Identity of Gessel and Stanton and the New Little Göllnitz Identities."
Advances in Applied Mathematics, 46 (1-4): 563-575.
doi: 10.1016/j.aam.2009.12.009 source: http://www.math.rutgers.edu/~asills/GesselStanton/GesselStanton.pdf