Further WZ-based methods for proving and generalizing Ramanujan's series
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Bibliographic record
Abstract
In 2002 and 2006, using a Wilf–Zeilberger-based method, Guillera introduced proofs for evaluations for what are considered as the simplest two series out of Ramanujan's 17 series for 1π. In this article, we show how the WZ method may be used in a fundamentally and nontrivially different way to prove these results, and to obtain identities for infinite families of Ramanujan-like series for 1π. We introduce a 3F2-recurrence that we had discovered experimentally, and we prove this recursion using the WZ method and apply it to obtain a series acceleration formula that we apply to formulate a new and simple proof for the Ramanujan series for 1π that has a convergence rate of 164, and we provide an infinite family of generalizations of this formula, and similarly for Ramanujan's series of convergence rate 14.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| 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) | 0.000 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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