On Decomposition of $\theta_2^{2n}(\tau)$ as the Sum of Lambert Series and Cusp forms

Dandan Chen; Rong Chen

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arXiv · 1910.09666

On Decomposition of $\theta_2^{2n}(\tau)$ as the Sum of Lambert Series and Cusp forms

Dandan Chen·Rong Chen

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Abstract

Based on the values of the Weierstrass elliptic function $\wp(z|\tau)$ at $z=\pi\tau/2$, $(\pi+\pi\tau)/{2}, (\pi+\pi\tau)/{4},(\pi+2\pi\tau)/{4}$ and the theory of modular forms on the arithmetic group $\Gamma_0(2)$, we decompose $\theta_2^{2n}(\tau)$ as sum of Eisenstein series and a cusp forms. Using the recurrence relation of $\wp^{(2n)}(z|\tau)$, we provide an algorithm to determine the exact form of these cusp forms.

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BibTeX RIS

Dandan Chen, Rong Chen. 2019-10-21. On Decomposition of $\theta_2^{2n}(\tau)$ as the Sum of Lambert Series and Cusp forms. https://arxiv.org/abs/1910.09666

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