# Conventions and verification sources for T557

Clarkson (2008), Section 5.1, equation (5.8), defines the generalized Hermite
family up to a nonzero constant as the Wronskian of consecutive physicists'
Hermite polynomials. Equations (5.6) and (5.7) fix that Hermite convention.
We use x for the source's z and make no variable scaling.
The Wronskian's leading coefficient is the product of the leading
coefficients 2^(m+j) times the Vandermonde determinant of the degrees m+j.
This gives 2^(mn+n(n-1)/2) times the product of j! for 1 <= j < n.
Division by that exact rational integer makes the stored polynomial monic.

Buckingham (2020), equations (3)–(7), uses a generalized Hermite normalization
with leading coefficient 2^(mn). Thus our polynomial is that paper's
H_(m,n)/2^(mn). In each differential recurrence the multiplicative factors on
both sides are equal, so the monic polynomials satisfy the same recurrence.
Equation (8) and the parameter assignment in (2) give the displayed Painlevé
IV solution; multiplicative constants disappear from its logarithmic derivative.

The independent construction in checks.py starts from the explicit Hermite
coefficient sum and iterates the n differential recurrence with exact
polynomial quotient-and-remainder tests. It shares no polynomial arithmetic
helper with the generator's Wronskian/Bareiss construction. Additional checks
use both differential recurrences, transposition symmetry, the fourth-order
bilinear equation (Clarkson, equation (5.9)), and Painlevé IV after clearing
denominators. Known controls and deliberately perturbed controls establish that
the checking code can distinguish correct from incorrect polynomials.

The m and n boundary axes are unit polynomials. Only positive indices are
stored; the unit extension is explicitly stated for boundary recurrences.
There is no claim that this family is an orthogonal-polynomial sequence.

Corpus screening used already_here, already_asked, Hermite/Painlevé/Okamoto
text searches, and api/lookup for x^2-1/2, x^2+1/2, x^4+3/4. No duplicate
family was found. Existing ingredient tables are T103 (Hermite_polynomials_H)
and T104 (Hermite_polynomials_He); the live search results supplied both slugs.
T556 is the companion Yablonskii–Vorob’ev draft and was not publicly accessible,
so a public companion cross-link is deferred. The proposed Painlevé equations
tag is absent from both live tag pages. It would connect T557, T556, T443 and
the proposed Umemura and generalized Okamoto tables; it is not created here.

Instruction provenance: agents/runs/20261027T185658Z-build-skills/manifest.json;
both archived skills SHA-256
 e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c.
The independent critique and the controller's offer remain outstanding.