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Siladić's theorem: Weighted words, refinement and companion


Author: Jehanne Dousse
Journal: Proc. Amer. Math. Soc. 145 (2017), 1997-2009
MSC (2010): Primary 05A17, 11P84
DOI: https://doi.org/10.1090/proc/13376
Published electronically: November 18, 2016
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Abstract: In a previous paper, the author gave a combinatorial proof and refinement of Siladić's theorem, a Rogers-Ramanujan type partition identity arising from the study of Lie algebras. Here we use the basic idea of the method of weighted words introduced by Alladi and Gordon to give a non-dilated version, further refinement and companion of Siladić's theorem. However, while in the work of Alladi and Gordon, identities were proved by doing transformations on generating functions, we use recurrences and $ q$-difference equations, as the original method seems difficult to apply in our case. As the non-dilated version features the same infinite product as Schur's theorem, another dilation allows us to find a new interesting companion of Schur's theorem, with difference conditions very different from the original ones.


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Jehanne Dousse
Affiliation: Institut für Mathematik, Universitat Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland

DOI: https://doi.org/10.1090/proc/13376
Received by editor(s): February 17, 2016
Received by editor(s) in revised form: July 12, 2016
Published electronically: November 18, 2016
Communicated by: Ken Ono
Article copyright: © Copyright 2016 American Mathematical Society