Where do the maximum absolute $q$-series coefficients of $(1-q)(1-q^2)(1-q^3)\dots(1-q^{n-1})(1-q^n)$ occur?

Alexander Berkovich; Ali K. Uncu

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arXiv · 1911.03707

Where do the maximum absolute $q$-series coefficients of $(1-q)(1-q^2)(1-q^3)\dots(1-q^{n-1})(1-q^n)$ occur?

Alexander Berkovich·Ali K. Uncu

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Abstract

We used the MACH2 supercomputer to study coefficients in the $q$-series expansion of $(1-q)(1-q^2)\dots(1-q^n)$, for all $n\leq 75000$. As a result, we were able to conjecture some periodic properties associated with the before unknown location of the maximum coefficient of these polynomials with odd $n$. Remarkably the observed period is 62,624.

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Alexander Berkovich, Ali K. Uncu. 2019-11-09. Where do the maximum absolute $q$-series coefficients of $(1-q)(1-q^2)(1-q^3)\dots(1-q^{n-1})(1-q^n)$ occur?. https://arxiv.org/abs/1911.03707

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