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arXiv · 2606.16213

On periodic sign changes for weighted representations of integers as colored sums of triangular and generalized pentagonal numbers

Ben Kane·Meenu Sharma
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Abstract

In this paper, we study sign changes and vanishing for the number of representations of an integer as a sum of triangular numbers plus three-colored sums of generalized pentagonal numbers with an even number of parts minus those with an odd number of parts. We study this by investigating coefficients appearing in the $q$-series expansion of $F(z) = \frac{\eta(z)}{{\eta(2z)}^2 {\eta(3z)}^3}$, where $\eta$ is the Dedekind-eta function. We use the Hardy-Ramanujan-Rademacher circle method to give an asymptotic formula for the coefficients.

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Ben Kane, Meenu Sharma. 2026-06-15. On periodic sign changes for weighted representations of integers as colored sums of triangular and generalized pentagonal numbers. https://arxiv.org/abs/2606.16213

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