Riccati equations and convolution formulae for functions of Rayleigh type
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Bibliographic record
Abstract
the Rayleigh functions σn(ν) = ∑ ∞ k=1 j −2n νk, n = 1, 2,..., where ±jνk are the (non-zero) zeros of the Bessel function Jν(z) and provided a convolution type sum formula for finding σn in terms of σ1,...,σn−1. His main tool was the recurrence relation for Bessel functions. Here we extend this result to a larger class of functions by using Riccati differential equations. We get new results for the zeros of certain combinations of Bessel functions and their first and second derivatives as well as recovering some results of Buchholz for zeros of confluent hypergeometric functions. 1
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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