Umemura polynomials for Painlevé III

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Polynomials

$n$

$S_n(x;m)$

x + m

x^3 + 3*x^2*m + 3*x*m^2 + m^3 - m

x^6 + 6*x^5*m + 15*x^4*m^2 + 20*x^3*m^3 - 5*x^3*m + 15*x^2*m^4 - 15*x^2*m^2 + 6*x*m^5 - 15*x*m^3 + 9*x*m + m^6 - 5*m^4 + 4*m^2

x^10 + 10*x^9*m + 45*x^8*m^2 + 120*x^7*m^3 - 15*x^7*m + 210*x^6*m^4 - 105*x^6*m^2 + 252*x^5*m^5 - 315*x^5*m^3 + 63*x^5*m + 210*x^4*m^6 - 525*x^4*m^4 + 315*x^4*m^2 + 120*x^3*m^7 - 525*x^3*m^5 + 630*x^3*m^3 - 225*x^3*m + 45*x^2*m^8 - 315*x^2*m^6 + 630*x^2*m^4 - 360*x^2*m^2 + 10*x*m^9 - 105*x*m^7 + 315*x*m^5 - 220*x*m^3 + m^10 - 15*m^8 + 63*m^6 - 85*m^4 + 36*m^2

x^15 + 15*x^14*m + 105*x^13*m^2 + 455*x^12*m^3 - 35*x^12*m + 1365*x^11*m^4 - 420*x^11*m^2 + 3003*x^10*m^5 - 2310*x^10*m^3 + 252*x^10*m + 5005*x^9*m^6 - 7700*x^9*m^4 + 2695*x^9*m^2 + 6435*x^8*m^7 - 17325*x^8*m^5 + 12915*x^8*m^3 - 2025*x^8*m + 6435*x^7*m^8 - 27720*x^7*m^6 + 36540*x^7*m^4 - 15255*x^7*m^2 + 5005*x^6*m^9 - 32340*x^6*m^7 + 67620*x^6*m^5 - 51310*x^6*m^3 + 11025*x^6*m + 3003*x^5*m^10 - 27720*x^5*m^8 + 85554*x^5*m^6 - 100905*x^5*m^4 + 40068*x^5*m^2 + 1365*x^4*m^11 - 17325*x^4*m^9 + 74970*x^4*m^7 - 127050*x^4*m^5 + 68040*x^4*m^3 + 455*x^3*m^12 - 7700*x^3*m^10 + 44940*x^3*m^8 - 104825*x^3*m^6 + 79730*x^3*m^4 - 12600*x^3*m^2 + 105*x^2*m^13 - 2310*x^2*m^11 + 17640*x^2*m^9 - 55230*x^2*m^7 + 66255*x^2*m^5 - 26460*x^2*m^3 + 15*x*m^14 - 420*x*m^12 + 4095*x*m^10 - 16935*x*m^8 + 31290*x*m^6 - 26145*x*m^4 + 8100*x*m^2 + m^15 - 35*m^13 + 427*m^11 - 2305*m^9 + 5768*m^7 - 6160*m^5 + 2304*m^3

Definition

The Umemura polynomials $S_n(x;m)$ associated with rational solutions of Painlevé III are defined by $S_{-1}=S_0=1$ and $S_{n+1}S_{n-1}=(x+m)S_n^2-x(S_nS_n^{\prime\prime}-(S_n^{\prime})^2)-S_nS_n^{\prime}$ for $n\geq0$, with derivatives taken with respect to $x$ [1], Definition 3.2.

Parameters

$n$

— Polynomial index ($n\geq0$)

Formulas

(1)

$S_n$ is monic in $x$, with $\deg_x S_n=n(n+1)/2$ for $n\geq0$ [2], Remark 2.4(3).

(2)

Define $p_k(x;m)$ by $(1+t)^m\exp(xt)=\sum_{k\geq0}p_k(x;m)t^k$, and set $p_k=0$ for $k<0$. Then $S_n(x;m)=\left(\prod_{j=1}^{n}(2j+1)^{n-j}\right)\det[p_{2j+1-i}(x;m)]_{i,j=0}^{n-1}$ [2], Theorem 2.5 and Remark 2.6(2).

(3)

$S_n(x;0)=x^{n(n+1)/2}$ and $S_n(x;1)=x^{n(n-1)/2}\theta_n(x)$, where $\theta_n(x)=x^n y_n(1/x)$ is the reverse of the Bessel polynomial $y_n$ [2], Remarks 2.6(3) and 2.6(4).

(4)

For $n\geq0$, let $w_n(x;m)=S_n(x;m-1)S_{n-1}(x;m)/(S_n(x;m)S_{n-1}(x;m-1))$. Then $w_n^{\prime\prime}=(w_n^{\prime})^2/w_n-w_n^{\prime}/x+(\alpha_n w_n^2+\beta_n)/x+w_n^3-1/w_n$, with $\alpha_n=2n+2m-1$ and $\beta_n=2n-2m+1$, as a rational-function identity [2], Theorem 2.3.

Comments

(5)

The letters $x$ and $m$ are independent polynomial indeterminates; $m$ denotes the parameter $\mu$ in [1] and [2]. The variable $x$ is the variable $z$ of [2], without rescaling. This indexing gives $S_0=1$ and $S_1=x+m$.

(6)

The determinant in (2) is the Wronskian of $p_1,p_3,\ldots,p_{2n-1}$, with rows in increasing derivative order. The empty determinant and empty product equal $1$.

Programs

(P1)

Python

# Run with SageMath's Python. The letter m denotes the parameter mu.
import numberdb.sage as numberdb
from sage.rings.integer_ring import ZZ
from sage.rings.rational_field import QQ
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing

R = PolynomialRing(QQ, names=('x', 'm'))
Z = PolynomialRing(ZZ, names=('x', 'm'))
x, m = R.gens()

def umemura(n):
    if n < 0 or n != int(n):
        raise ValueError('n must be a nonnegative integer')
    previous = current = R.one()
    for k in range(int(n)):
        d = current.derivative(x)
        numerator = (x+m)*current**2 - x*(current*d.derivative(x)-d**2) - current*d
        following, remainder = numerator.quo_rem(previous)
        assert remainder == 0
        previous, current = current, following
    return Z(current)

polynomials = {n: umemura(n) for n in range(6)}
umemura(6)  # Compute the next polynomial with the same normalization.

References

[1]

Peter A. Clarkson, Classical solutions of the degenerate fifth Painlevé equation, Journal of Physics A: Mathematical and Theoretical 56 (2023), 134002. (doi)

[2]

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. (arXiv) (doi)

Links

[3]

Clarkson: Classical solutions of the degenerate fifth Painlevé equation (full text)

[4]

Clarkson, Law and Lin: A Constructive Proof for the Umemura Polynomials of the Third Painlevé Equation (full text)

Similar tables

Bessel polynomials — give the specialization at $m=1$ after reversing the polynomial and multiplying by $x^{n(n-1)/2}$

Yablonskii–Vorobev polynomials — are the corresponding polynomial family for rational solutions of Painlevé II

Generalized Hermite polynomials for Painlevé IV — form another Wronskian family associated with rational Painlevé solutions

Data properties

Entries are of type: integral polynomial

How well the digits are known: exact

Table is complete: no (it holds every index $0\leq n\leq5$, retaining the full symbolic dependence on $x$ and $m$. This initial segment covers every intervening polynomial through degree $15$; the longest expanded entry has $898$ characters, while $S_6$ has $2014$. The cutoff is a practical readability choice for low-order reference polynomials)

How they were obtained:

Computed exactly in $\mathbb{Z}[x,m]$ by the defining recurrence, with a zero-remainder check at every monic division. All six entries were independently reconstructed from the coefficient expansion of $(1+t)^m\exp(xt)$ and its Wronskian determinant, evaluated by the Leibniz formula over $\mathbb{Q}[x,m]$.

The five explicit polynomials in [2], Table 1, agree coefficient by coefficient. Degree, monicity, integrality, parity under simultaneous sign reversal, the $m=0$ and $m=1$ specializations, the recurrence, the parameter-shift identity in [1], Remark 5.4, and the rational Painlevé III identity were checked exactly for every stored index. The recurrence and parameter-shift checks at $n=5$ use an independently computed $S_6$. Controls include known nonzero and zero determinants and deliberately altered polynomials. The independent checks were repeated on values read back from the database. The program agrees at every stored index and at $n=6$. All coefficients are exact, with no numerical argument rounding.