Ramanujan--Fine integrals for level 10

Shaun Cooper; Timothy Huber; Jeffery Opoku

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arXiv · 2410.19186

Ramanujan--Fine integrals for level 10

Shaun Cooper·Timothy Huber·Jeffery Opoku

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Abstract

We investigate the question of when an eta quotient is a derivative of a formal power series with integer coefficients and present an analysis in the case of level 10. As a consequence, we establish and classify an infinite number of integral evaluations such as $$ \int_0^{e^{-2\pi/\sqrt{10}}} q\prod_{j=1}^\infty \frac{(1-q^j)^3(1-q^{10j})^8}{(1-q^{5j})^7} \text{d} q = \frac14\left(\sqrt{10-4\sqrt{5}}-1\right). $$ We describe how the results were found and give reasons for why it is reasonable to conjecture that the list is complete for level 10.

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Shaun Cooper, Timothy Huber, Jeffery Opoku. 2024-10-24. Ramanujan--Fine integrals for level 10. https://arxiv.org/abs/2410.19186

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