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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On some polynomials and series of Bloch–Pólya type
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by Alexander Berkovich and Ali Kemal Uncu PDF
Proc. Amer. Math. Soc. 146 (2018), 2827-2838 Request permission


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
  • MR Author ID: 247760
  • Email:
  • Ali Kemal Uncu
  • Affiliation: Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Altenberger Straße 69, A-4040 Linz, Austria
  • MR Author ID: 1129887
  • ORCID: 0000-0001-5631-6424
  • Email:
  • 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
  • © Copyright 2018 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 146 (2018), 2827-2838
  • MSC (2010): Primary 05A17, 11B65; Secondary 05A19, 05A30, 11P81
  • DOI:
  • MathSciNet review: 3787346