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On some polynomials and series of Bloch-Pólya type


Authors: Alexander Berkovich and Ali Kemal Uncu
Journal: Proc. Amer. Math. Soc. 146 (2018), 2827-2838
MSC (2010): Primary 05A17, 11B65; Secondary 05A19, 05A30, 11P81
DOI: https://doi.org/10.1090/proc/13982
Published electronically: March 9, 2018
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Abstract: We will show that $ (1-q)(1-q^2)\dots (1-q^m)$ is a polynomial in $ q$ with coefficients from $ \{-1,0,1\}$ iff $ m=1,\ 2,\ 3,$ or 5 and explore some interesting consequences of this result. We find explicit formulas for the $ q$-series coefficients of $ (1-q^2)(1-q^3)(1-q^4)(1-q^5)\dots $ and $ (1-q^3)(1-q^4)(1-q^5)(1-q^6)\dots $. In doing so, we extend certain observations made by Sudler in 1964. We also discuss the classification of the products $ (1-q)(1-q^2)\dots (1-q^m)$ and some related series with respect to their absolute largest coefficients.


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Additional Information

Alexander Berkovich
Affiliation: Department of Mathematics, University of Florida, 358 Little Hall, Gainesville, Florida 32611
Email: alexb@ufl.edu

Ali Kemal Uncu
Affiliation: Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Altenberger Straße 69, A-4040 Linz, Austria
Email: akuncu@risc.jku.at

DOI: https://doi.org/10.1090/proc/13982
Keywords: Pentagonal numbers, Bloch--P\'olya type series, $q$-series identities, $q$-binomial theorem, partition theorems
Received by editor(s): June 9, 2017
Received by editor(s) in revised form: October 9, 2017
Published electronically: March 9, 2018
Additional Notes: Research of the first author was partly supported by the Simons Foundation, award ID: 308929.
Research of the second author was supported by the Austrian Science Fund (FWF): SFB F50-07.
Communicated by: Ken Ono
Article copyright: © Copyright 2018 American Mathematical Society

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