On some polynomials and series of Bloch–Pólya type

Berkovich, Alexander; Uncu, Ali

doi:10.1090/proc/13982

On some polynomials and series of Bloch–Pólya type
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by Alexander Berkovich and Ali Kemal Uncu

Proc. Amer. Math. Soc. 146 (2018), 2827-2838

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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