Ramanujan--Fine integrals for level 10
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908277335392256 |
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| author | Cooper, Shaun Huber, Timothy Opoku, Jeffery |
| author_facet | Cooper, Shaun Huber, Timothy Opoku, Jeffery |
| contents | 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 $$ \int_0^{e^{-2π/\sqrt{10}}} q\prod_{j=1}^\infty \frac{(1-q^j)^3(1-q^{10j})^8}{(1-q^{5j})^7} \text{d} q = \frac14\left(\sqrt{10-4\sqrt{5}}-1\right). $$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_19186 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ramanujan--Fine integrals for level 10 Cooper, Shaun Huber, Timothy Opoku, Jeffery Number Theory 11F11, 33E05 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 $$ \int_0^{e^{-2π/\sqrt{10}}} q\prod_{j=1}^\infty \frac{(1-q^j)^3(1-q^{10j})^8}{(1-q^{5j})^7} \text{d} q = \frac14\left(\sqrt{10-4\sqrt{5}}-1\right). $$ 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. |
| title | Ramanujan--Fine integrals for level 10 |
| topic | Number Theory 11F11, 33E05 |
| url | https://arxiv.org/abs/2410.19186 |