Nested radicals obtained via the Wilf--Zeilberger method and related results
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Bibliographic record
Abstract
We apply the Maple implementation of the Wilf--Zeilberger method to prove an acceleration formula for a family of ₃F₂(1)-series with three free parameters, so as to accelerate the convergence rate from 1 to ¼ for such series. We apply our series transform to determine new, fast-converging, rational, hypergeometric series for constants of the form √{2 ± √{2 + √{2 + ⋯ √{2}}}}, which are of significance in classical geometry. Our series of convergence rate ¼ closely resemble Ramanujan's series for 1/π of the same convergence rate. Our acceleration for the family of ₃F₂(1)-series considered in this article is based on a WZ pair generated by Maple and associated with a special case of Gauss's ₂F₁(1)-identity; a corresponding case of Dixon's ₃F₂(1)-identity may be used to further demonstrate the applicability of our telescoping-based approach to series acceleration, via the new series for π, π², Apéry's constant ζ(3), and ∛2 that we introduce.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
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 | 0.001 |
| 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 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it