## 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

## 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.## References

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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: alexb@ufl.edu
**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: akuncu@risc.jku.at
- 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: https://doi.org/10.1090/proc/13982
- MathSciNet review: 3787346