Sums of the even integral powers of the cosecant and secant
Bibliographic record
Abstract
Special finite sums of the even powers P of the cosecant and of the secant are studied,∑k csc2m(kπ/N) and ∑k sec2m (kπ/N), with positive integers N ≥ 3, m and 1 ≥ k ≥ N/2 . The main result of this article is that these power sums are even polynomials in N, of order 2m, whose coefficients are rational. The approach is based on new differential identities for the functions csc2z and sec2z. The Mittag-Leffler expansions for these functions are invoked and the corresponding infinite series are summed to give closed form expressions for the desired sums. Specific polynomial coefficients are obtained, for 1 ≤ m ≤ 6 and for all N ≥ 3, to illustrate the method. Similar sums involving the cotangent and the tangent are also examined.
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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 | 0.003 |
| 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 | 0.002 | 0.003 |
| 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".