Nested radicals obtained via the Wilf--Zeilberger method and related results
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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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".