Applications via series accelerations of new identities involving Catalan-type numbers
Bibliographic record
Abstract
We introduce infinite families of terminating hypergeometric identities involving generalizations of Catalan numbers, generalizing results introduced by Chu and K\i l\i\c{c}, and we apply Wilf—Zeilberger (WZ) pairs associated with our new identities via a series acceleration method. We apply a WZ pair introduced in our article to prove an identity for accelerating the convergence for a family of ${}_{3}F_{2}(1)$-series with three real parameters from $1$ to $\frac{1}{4}$, and we apply this identity to generalize Ramanujan-like series for $\frac{1}{\pi}$, $\frac{\sqrt{2}}{\pi}$, $\frac{\sqrt{3}}{\pi}$, and $ \frac{\sqrt{2 \pm \sqrt{2}}}{\pi } $ that are due to Chu et al. A fast-converging series for $\pi^2$ due to Guillera is also a special case of our acceleration identity. We also apply another WZ pair introduced in this article to prove an identity for accelerating the convergence of a ${}_{3}F_{2}(1)$-family with three real parameters from $1$ to $\frac{1}{16}$, and we apply this result, via a series bisection, to formulate a new WZ proof of Ramanujan's series for $\frac{1}{\pi}$ of convergence rate $\frac{1}{4}$. A number of our finite sums involving Catalan-type numbers are such that up-to-date versions of the Maple Computer Algebra System cannot compute WZ pairs for such sums, which is representative of the computationally challenging nature of our results.
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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 | 0.002 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.004 | 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".