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3272 bytes, as of the version from 2026-10-28 08:16 (current). Recorded here, not run.
# Checked sources and conventions
The build used the skill versions in the manifest
`../20261027T182939Z-build-skills/manifest.json` (both SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`).
Sources were read through the web tool on 2026-10-27. A direct urllib download
of the arXiv PDF returned HTTP 403, so no local PDF archive is claimed. The
web tool successfully supplied the full PDF text.
1. NIST DLMF, https://dlmf.nist.gov/32.8.ii , equations 32.8.5–32.8.10.
The recurrence uses Q_0=1, Q_1=x and the coefficient -4 in front of
Q Q''-(Q')^2. Explicit Q_2 through Q_6 provide external initial values.
The logarithmic derivative Q_(n-1)'/Q_(n-1)-Q_n'/Q_n gives the rational
Painlevé II solution. The Wronskian uses odd-index coefficients from
exp(x*t-4*t^3/3); it requires normalization to make the result monic.
2. Marco Bertola and Thomas Bothner, arXiv:1401.1408,
https://arxiv.org/abs/1401.1408 and https://arxiv.org/pdf/1401.1408,
page 1, equations (1.1)–(1.4). The explicit normalized Jacobi–Trudi
determinant is product_(k=1)^n (2k+1)^(n-k) times
det[p_(n-2i+j)(x)]_(i,j=0)^(n-1), where p_r is the coefficient of t^r
in exp(x*t-4*t^3/3), with p_r=0 for r<0. This is the independent exact
construction used in checks.py. The paper states degree n(n+1)/2,
integrality, monicity, and w(x;-n)=-w(x;n).
No disagreement about the defining recurrence was found between these sources
and the family proposal. An explicit-value discrepancy is resolved as follows. The choice of n<=12 was made from the length study, not
inherited as a required grid. The title follows issue #223; Keywords include
the ASCII and reversed-order name variants. No issue provenance is inserted
into mathematical table prose.
## Resolved discrepancy in the retrieved handbook text
The text returned for DLMF 32.8.7 gave the coefficient of x^3 in Q_5 as
-313600. The source-comparison test failed at precisely this coefficient.
Both the exact differential recurrence and Bertola–Bothner determinant give
-3136000. Replacing it with -313600 yields recurrence residual
2822400*x^3*Q_3 and fails Painlevé II. Thus this is an erroneous printed or
extracted coefficient, not a different convention. Direct DLMF TeX and HTML
downloads were blocked with HTTP 403, so we do not claim to have determined
whether the error is in the live typography or its retrieved representation.
The publisher-hosted paper by Pieter Roffelsen, *Irrationality of the Roots
of the Yablonskii–Vorob’ev Polynomials and Relations between Them*, SIGMA 6
(2010), 095, Table 1 (page 2), gives -3136000 and complete Q_2 through Q_8.
URL: https://sigma-journal.com/2010/095/sigma10-095.pdf ;
DOI: 10.3842/SIGMA.2010.095; arXiv:1012.2933. The web tool supplied the PDF
text; its screenshot fetch failed, and direct PDF download returned HTTP 406.
The final independent reference comparison uses that paper's Q_0 through Q_8.
All source failures are retained in source-records.json.
A second checker failure was our own transcription of the Q_8 constant: three
extra zeros had been appended to -991048439693312000000. The paper and
generator agreed; the copied reference literal was corrected. No stored
values were involved in either failure.