A note on Bernoulli identities associated with the Weierstrass ℘-function
Bibliographic record
Abstract
Various types of Bernoulli identities have been discussed by Euler, Ramanujan, Rademacher, Eie, and others. In particular, by using certain identities involving zeta functions, Eie constructed several Bernoulli identities systematically. There are many expressions for the Bernoulli number B 2k , which express B 2k as a sum of the products of lower-order Bernoulli numbers in two terms or in three terms. In this article, we show that there is no unique method to calculate the Bernoulli number B 2k as a sum of the products of lower-order Bernoulli numbers in three terms. By employing a technique which is substantially different from the method of Eie’s proof, we derive several Bernoulli identities through the properties of the Weierstrass ℘-function. We also discuss the relationship between these and other known Bernoulli identities.
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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 | 0.003 | 0.005 |
| Scholarly communication | 0.002 | 0.004 |
| 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".