# On some polynomials and series of Bloch-Polya Type

## TL;DR

This paper characterizes when certain polynomial products have coefficients only from {-1,0,1}, finds explicit formulas for related q-series, and extends previous observations on their coefficient behavior.

## Contribution

It provides a complete classification of when the polynomial products have coefficients in {-1,0,1} and derives explicit formulas for specific q-series coefficients.

## Key findings

- Polynomial products have coefficients in {-1,0,1} iff m=1,2,3, or 5.
- Explicit formulas for coefficients of certain q-series.
- Extended observations on the largest coefficients of related series.

## 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.