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arXiv · 1705.07504

On some polynomials and series of Bloch-Polya Type

Alexander Berkovich·Ali K. Uncu
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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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Alexander Berkovich, Ali K. Uncu. 2017-10-11. On some polynomials and series of Bloch-Polya Type. https://arxiv.org/abs/1705.07504

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