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We present a new companion to the deep partition theorem of Göllnitz and discuss it in the context of a generalization of Göllnitz’s theorem by Alladi–Andrews–Gordon that was obtained by the method of weighted words. After providing a q-theoretic proof of the new companion theorem, we discuss its analytic representation and its link to the key identity of Alladi–Andrews–Gordon.
A key identity in three free parameters involving partitions into distinct parts is proved using Jackson'sq-analog of Dougall's summation. This identity is shown to be combinatorially equivalent to a reformulation of a deep partition theorem of Göllnitz obtained by the use of a quartic transformation.
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