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\iteman{io-port 05895464}
\itemau{Savage, Carla D.; Sills, Andrew V.}
\itemti{On an identity of Gessel and Stanton and the new little G\"ollnitz identities.}
\itemso{Adv. Appl. Math. 46, No. 1-4, 563-575 (2011).}
\itemab
Summary: We show that an identity of Gessel and Stanton [{\it I. Gessel} and {\it D. Stanton}, ``Applications of $q$-Lagrange inversion to basic hypergeometric series,'' Trans. Am. Math. Soc. 277, 173-201 (1983; Zbl 0513.33001), Equation (7.24)] can be viewed as a symmetric version of a recent analytic variation of the little G\"ollnitz identities. This is significant, since the G\"ollnitz-Gordon identities are considered the usual symmetric counterpart to little G\"ollnitz theorems. Is it possible, then, that the Gessel-Stanton identity is part of an infinite family of identities like those of G\"ollnitz-Gordon? Toward this end, we derive partners and generalizations of the Gessel-Stanton identity. We show that the new little G\"ollnitz 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.
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\itemut{integer partitions; $q$-series identities; $q$-Gauss summation; little G\"ollnitz partition theorems; G\"ollnitz-Gordon partition theorem; Lebesgue identity}
\itemli{doi:10.1016/j.aam.2009.12.009}
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