# Argument selection for T558: Umemura polynomials for Painlevé III

The only free table parameter is the integer polynomial index n. Its role is
`discrete_index`: the n-th member of the monic polynomial sequence associated
with rational Painlevé III solutions. It is not a sampled continuous coupling.
The family is infinite, n >= 0. The auxiliary initial condition S_{-1}=1 is
part of the recurrence, not a second copy of the unit polynomial in the table.

Both x and mu are polynomial indeterminates in ZZ[x,mu]. Keeping mu symbolic
covers the parameter dependence exactly; there is no selected grid of mu or x,
and no hidden numerical argument. The variable order is x, mu. The convention
is Clarkson, Law and Lin (2023), equation (1.5) and Theorem 2.3: S_0=1,
S_1=x+mu, and degree in x equal to n(n+1)/2. The paper's z is renamed x
without rescaling. Older indexing with two initial rows S_0=S_1=1 is not used.

The default enumeration is every n from 0 through 5, in increasing order.
The six entries include all intervening indices and reach degree 15. This
initial reference segment serves identification of the low-order polynomial
factors in rational Painlevé III solutions, with the full symbolic coupling
retained. Each member need not have a separate application. The finite bound
is a noncanonical readability choice, not a mathematical end of the family.

Measured expanded lengths at n=0,1,2,3,4,5,6 are respectively
1, 6, 37, 138, 390, 959 and 2129 characters. The selected six entries occupy
about 1.8 KB including entry overhead in the shared measure() check. The
next polynomial more than doubles the longest entry. No size quota motivates
the cutoff. This is one indexed family, so breadth means retaining the whole
initial segment and the full two-variable objects, rather than deepening a
single numerical specialization such as mu=0 or mu=1. Six entries suffice for
that bounded scope; no artificial minimum or arbitrary extra rows are imposed.

Checked sources:
- P. A. Clarkson, C.-K. Law and C.-H. Lin, A Constructive Proof for the Umemura
  Polynomials of the Third Painlevé Equation, SIGMA 19 (2023), 080.
  https://doi.org/10.3842/SIGMA.2023.080 . Equation (1.5), Theorems 2.3 and 2.5,
  Remarks 2.4(3) and 2.6, and Table 1 establish the recurrence, indexing,
  normalization, determinant representation and printed initial members.
- K. Kajiwara and T. Masuda, On the Umemura Polynomials for the Painlevé III
  Equation, Physics Letters A 260 (1999), 462–467.
  https://arxiv.org/abs/solv-int/9903015 . Theorem 1 and equations (12)–(16)
  give the staircase determinant and its coefficient generating function.

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