Elliptic integral representation of Bessel moments
Bibliographic record
Abstract
We record and substantially extend what is known about the closed forms for various Bessel function moments arising in quantum field theory, condensed matter theory and other parts of mathematical physics. In particular, we develop formulae for integrals of products of six or fewer Bessel functions. In consequence, we are able to discover and prove closed forms for cn,k := ∫∞₀ tkKⁿ₀(t) dt with integers n = 1, 2, 3, 4 and k ≽ 0, obtaining new results for the even moments c₃,₂k and c₄,₂k. We also derive new closed forms for the odd moments sn,₂k₊₁ := ∫∞₀ t²k⁺¹I₀(t)Kⁿ₀¹₀(t) dt with n = 3, 4 and for tn,₂k₊₁ := ∫∞ ₀ t²k⁺¹I²₀(t)Kⁿ⁻²₀(t)dt with n = 5, relating the latter to Green functions on hexagonal, diamond and cubic lattices. We conjecture the values of s₅,₂k₊₁, make substantial progress on the evaluation of c₅,₂k₊₁, s₆,₂k₊₁ and t₆,₂k₊₁ and report more limited progress regarding c₅,₂k, c₆,₂k₊₁ and c₆,₂k. In the process, we obtain eight conjectural evaluations, each of which has been checked to 1200 decimal places. One of these lies deep in four-dimensional quantum field theory and two are probably provable by delicate combinatorics. There remains a hard core of five conjectures whose proofs would be most instructive, to mathematicians and physicists alike.
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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 | 0.004 |
| 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 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 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".