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{
"Title": "Umemura polynomials for Painlevé III",
"Definition": "The Umemura polynomials $S_n(x;\\mu)\\in\\mathbb{Z}[x,\\mu]$ for Painlevé III are determined by $S_{-1}=S_0=1$ and $S_{n+1}S_{n-1}=(x+\\mu)S_n^2-x\\bigl(S_nS_n^{\\prime\\prime}-(S_n^\\prime)^2\\bigr)-S_nS_n^\\prime$ for $n\\geq0$, with derivatives taken in $x$ CITE{CLL}.",
"Parameters": {
"n": {
"type": "Z",
"display": "$n$",
"title": "polynomial index",
"constraints": "$n\\geq0$"
}
},
"Comments": {
"comment-convention": "The polynomials are monic in $x$. Both $x$ and $\\mu$ are indeterminates, so each entry retains the full dependence on $\\mu$. The indexing is that of CITE{CLL}, Theorem 2.3, with $S_0=1$ and $S_1=x+\\mu$; its variable $z$ is renamed $x$ without rescaling.",
"comment-solutions": "The quotient in CITE{formula-painleve} gives rational solutions of the third Painlevé equation. The fixed equation normalization is $\\gamma=1$, $\\delta=-1$ in the usual four-parameter form CITE{CLL}, equation (1.2)."
},
"Formulas": {
"formula-degree": "$\\deg_x S_n=n(n+1)/2$ for $n\\geq0$ CITE{CLL}, Remark 2.4(3).",
"formula-determinant": "Define $p_k(x;\\mu)$ by $\\sum_{k\\geq0}p_k(x;\\mu)t^k=(1+t)^\\mu e^{xt}$, and put $p_k=0$ for $k<0$. Then $S_n(x;\\mu)=\\left(\\prod_{j=1}^{n}(2j+1)^{n-j}\\right)\\det\\left[p_{2j-i}(x;\\mu)\\right]_{i,j=1}^{n}$, with the empty determinant and empty product equal to $1$ CITE{CLL}, Theorem 2.5 and Remark 2.6(2), CITE{KM}.",
"formula-specializations": "$S_n(x;0)=x^{n(n+1)/2}$ and $S_n(x;1)=x^{n(n-1)/2}\\theta_n(x)$, where $\\theta_n(x)=x^ny_n(1/x)$ is the reverse of the HREF{Bessel_polynomials}[Bessel polynomial $y_n$] CITE{CLL}, Remarks 2.6(3) and 2.6(4).",
"formula-painleve": "For $n\\geq0$, let $w_n=\\frac{S_n(x;\\mu-1)S_{n-1}(x;\\mu)}{S_n(x;\\mu)S_{n-1}(x;\\mu-1)}=1+\\frac{d}{dx}\\log\\frac{S_{n-1}(x;\\mu-1)}{S_n(x;\\mu)}$. Then $w_n^{\\prime\\prime}=\\frac{(w_n^\\prime)^2}{w_n}-\\frac{w_n^\\prime}{x}+\\frac{\\alpha_n w_n^2+\\beta_n}{x}+w_n^3-\\frac1{w_n}$, where $\\alpha_n=2n+2\\mu-1$ and $\\beta_n=2n-2\\mu+1$, as a rational-function identity CITE{CLL}, Theorem 2.3."
},
"Similar tables": [
{
"table": "HREF{Bessel_polynomials}[Bessel polynomials]",
"relation": "their reversals occur in the specialization $\\mu=1$"
},
{
"table": "HREF{T556}[Yablonskii–Vorob’ev polynomials]",
"relation": "give polynomial factors for rational solutions of Painlevé II"
},
{
"table": "HREF{T557}[Generalized Hermite polynomials for Painlevé IV]",
"relation": "give a different polynomial construction for rational solutions of Painlevé IV"
}
],
"Tags": [
"polynomial",
"special functions",
"recurrence"
],
"Keywords": [
"Umemura",
"Painleve III",
"Painlevé III",
"rational solutions"
],
"Links": {
"KM": {
"title": "Kajiwara and Masuda: On the Umemura Polynomials for the Painlevé III equation",
"url": "https://arxiv.org/abs/solv-int/9903015"
}
},
"References": {
"CLL": {
"bib": "Peter A. Clarkson, Chun-Kong Law and Chia-Hua Lin, A Constructive Proof for the Umemura Polynomials of the Third Painlevé Equation, SIGMA 19 (2023), 080, 20 pages.",
"doi": "10.3842/SIGMA.2023.080",
"arxiv": "1609.00495"
}
},
"Programs": {
"program-python": {
"language": "Python",
"code": "import numberdb.sage as numberdb\nfrom sage.rings.integer_ring import ZZ\nfrom sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing\n\nR = PolynomialRing(ZZ, names=('x', 'mu'), order='lex')\nx, mu = R.gens()\n_POLYS = [R.one(), x + mu]\n\n\ndef monic_quotient(numerator, denominator):\n \"\"\"Exact division in x; a nonzero remainder is a hard failure.\"\"\"\n degree = denominator.degree(x)\n assert denominator.coefficient({x: degree}) == 1\n quotient = R.zero()\n remainder = numerator\n while remainder and remainder.degree(x) >= degree:\n top = remainder.degree(x)\n term = remainder.coefficient({x: top}) * x**(top - degree)\n quotient += term\n remainder -= term * denominator\n assert remainder == 0, 'Nonzero recurrence remainder'\n return quotient\n\n\ndef polynomial(n):\n n = int(n)\n if n < 0:\n raise ValueError('n must be nonnegative')\n while len(_POLYS) <= n:\n previous, current = _POLYS[-2:]\n derivative = current.derivative(x)\n numerator = ((x + mu)*current**2\n - x*(current*derivative.derivative(x) - derivative**2)\n - current*derivative)\n _POLYS.append(monic_quotient(numerator, previous))\n return _POLYS[n]\n\npolynomials = {n: polynomial(n) for n in range(6)}\n"
}
},
"Display properties": {
"number-header": "$S_n(x;\\mu)$"
},
"Data properties": {
"type": "Z[]",
"rigour": "exact",
"complete": "no",
"complete-note": "it holds every index $0\\leq n\\leq5$ in this discrete family of polynomials for rational Painlevé III solutions, retaining the full symbolic dependence on $\\mu$ rather than sampling it. This initial reference segment includes every intervening index through degree $15$; the noncanonical cutoff is a readability choice, with $959$ characters in the largest expanded entry before $S_6$ reaches $2129$",
"rigour details": "Computed over $\\mathbb{Z}[x,\\mu]$ by the defining differential recurrence using monic polynomial long division with a zero-remainder assertion. All six entries were independently reconstructed over $\\mathbb{Q}[x,\\mu]$ from the coefficient sum for $(1+t)^\\mu e^{xt}$ and a permutation expansion of the Wronskian determinant. The five explicit polynomials in CITE{CLL}, Table 1, agreed coefficient by coefficient. Degree, monicity, integer coefficients, both specializations in CITE{formula-specializations}, the recurrence, the logarithmic-derivative identity, and the Painlevé III equation after clearing denominators were checked for every stored index. The checks include known nonzero and zero determinant controls and deliberately perturbed polynomials. These checks were performed locally. The server refused the attempted fill because its polynomial parser does not accept the variable name mu, so no database readback or generator verify was possible."
},
"Numbers": []
}