On a higher-order version of a formula due to Ramanujan
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Bibliographic record
Abstract
A special case of an Entry in Part II of Ramanujan's Notebooks is such that 1+15(12)2+19(1⋅32⋅4)2+⋯=Γ4(14)16π2.This formula leads us to consider the higher-order version of the above series given by replacing the squares of normalized central binomial coefficients with fourth powers. Using a Fourier–Legendre expansion introduced in a 2022 article by Cantarini, together with a multiple elliptic integral evaluation conjectured by Wan and proved by Zhou, we prove the very natural extension shown below of Ramanujan's formula: 1+15(12)4+19(1⋅32⋅4)4+⋯=Γ8(14)96π5.Furthermore, and in a closely related way, we show how a main result in an article by Papanikolas et al. concerning a Calabi–Yau threefold is equivalent to the evaluation of a Clebsch–Gordan-type multiple elliptic integral related to the work of Zhou and Brychkov.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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