Yablonskii–Vorobev polynomials

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Polynomials

$n$

$Q_n(x)$

x^3 + 4

x^6 + 20*x^3 - 80

x^10 + 60*x^7 + 11200*x

x^15 + 140*x^12 + 2800*x^9 + 78400*x^6 - 3136000*x^3 - 6272000

x^21 + 280*x^18 + 18480*x^15 + 627200*x^12 - 17248000*x^9 + 1448832000*x^6 + 19317760000*x^3 - 38635520000

x^28 + 504*x^25 + 75600*x^22 + 5174400*x^19 + 62092800*x^16 + 13039488000*x^13 - 828731904000*x^10 - 49723914240000*x^7 - 3093932441600000*x

x^36 + 840*x^33 + 240240*x^30 + 32771200*x^27 + 2018016000*x^24 + 124309785600*x^21 - 6629855232000*x^18 + 407736096768000*x^15 + 126696533483520000*x^12 + 1769729356595200000*x^9 + 37164316488499200000*x^6 - 743286329769984000000*x^3 - 991048439693312000000

x^45 + 1320*x^42 + 646800*x^39 + 160160000*x^36 + 21525504000*x^33 + 1876485811200*x^30 + 46132742656000*x^27 + 8403341506560000*x^24 + 483515342069760000*x^21 - 318108851847987200000*x^18 - 18430315465453731840000*x^15 - 558505348189165977600000*x^12 + 9636955027577765888000000*x^9 - 404752111158266167296000000*x^6 - 4047521111582661672960000000*x^3 + 5396694815443548897280000000

10:

x^55 + 1980*x^52 + 1544400*x^49 + 639038400*x^46 + 155387232000*x^43 + 24080581324800*x^40 + 2227039406592000*x^37 + 182934280488960000*x^34 + 4351638078627840000*x^31 - 2587344319342182400000*x^28 + 716670308138534830080000*x^25 + 136012478911949832192000000*x^22 + 9157516514955772035072000000*x^19 + 56029256530337130872832000000*x^16 + 11766143871370797483294720000000*x^13 - 299121524196626496019759104000000*x^10 - 17947291451797589761185546240000000*x^7 - 558360178500369459236883660800000000*x

11:

x^66 + 2860*x^63 + 3363360*x^60 + 2179777600*x^57 + 870661792000*x^54 + 228783281126400*x^51 + 40292945172480000*x^48 + 5092086356797440000*x^45 + 406843623541555200000*x^42 + 14155180258726707200000*x^39 + 10054694231886601256960000*x^36 - 807997665873097693593600000*x^33 - 823341031180521544980889600000*x^30 - 130803412106545939641008128000000*x^27 - 6431598357117399252892385280000000*x^24 - 382362771215916269483543494656000000*x^21 + 8922795066785368376269414072320000000*x^18 + 92751602222889943885806902968320000000*x^15 - 96126729970445105732762654159667200000000*x^12 - 1031603931390142598107696776347648000000000*x^9 - 17330946047354395648209305842640486400000000*x^6 + 231079280631391941976124077901873152000000000*x^3 + 231079280631391941976124077901873152000000000

12:

x^78 + 4004*x^75 + 6806800*x^72 + 6569763200*x^69 + 4039754118400*x^66 + 1685413778048000*x^63 + 493466478529024000*x^60 + 104481874400583680000*x^57 + 15961539852772433920000*x^54 + 1785957180671751127040000*x^51 + 205915602778791313408000000*x^48 - 11564119189184390692864000000*x^45 - 4446877560456380103262208000000*x^42 + 3888698098542152937755377664000000*x^39 + 1567871436413636394499428057088000000*x^36 + 206243284150088848325534485053440000000*x^33 + 17696462812804866520922942938808320000000*x^30 + 376584387191908573557471277010124800000000*x^27 + 23103908827042868966710689491058688000000000*x^24 + 4603016741862816272756543016063205376000000000*x^21 - 558103916505655486431852602184735653888000000000*x^18 - 28938454449378866708427691631213692321792000000000*x^15 - 781093028993827885307984727503522119024640000000000*x^12 + 9819455221636693415300379431472849496309760000000000*x^9 - 274944746205827415628410624081239785896673280000000000*x^6 - 2199557969646619325027284992649918287173386240000000000*x^3 + 2199557969646619325027284992649918287173386240000000000

Definition

The Yablonskii–Vorobev polynomials $Q_n(x)$ are determined by $Q_0(x)=1$, $Q_1(x)=x$ and $Q_{n+1}(x)Q_{n-1}(x)=xQ_n(x)^2-4\bigl(Q_n(x)Q_n^{\prime\prime}(x)-(Q_n^{\prime}(x))^2\bigr)$ for $n\geq1$, with derivatives taken in the indeterminate $x$, [3].

Parameters

$n$

— index ($n\geq0$)

Formulas

(1)

$\deg Q_n=n(n+1)/2$ for $n\geq0$ [1].

(2)

For $n\geq1$, let $w_n(x)=Q_{n-1}^{\prime}(x)/Q_{n-1}(x)-Q_n^{\prime}(x)/Q_n(x)$. Then $w_n^{\prime\prime}=2w_n^3+xw_n+n$ [2], Theorem 1.

(3)

For $m\geq0$, define $p_m(x)$ by $\sum_{m=0}^{\infty}p_m(x)t^m=\exp(xt-4t^3/3)$, and set $p_m(x)=0$ for $m<0$. For $n\geq1$, $Q_n(x)=\left(\prod_{k=1}^{n}(2k+1)^{n-k}\right)\det\left[p_{n-2i+j}(x)\right]_{i,j=0}^{n-1}$ [1], equations (1.3) and (1.4).

Comments

(4)

The polynomials are monic in $x$, with integer coefficients. The indeterminate $x$ has the scaling used in [3], equation (32.8.6), and [1], Section 1.

(5)

The logarithmic derivatives in (2) give the rational solutions of the second Painlevé equation at positive integer parameter $n$ [2], Theorem 1.

Programs

(P1)

Python

import numberdb.sage as numberdb
from sage.rings.integer_ring import ZZ
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing

R = PolynomialRing(ZZ, 'x')
x = R.gen()
Q = [R.one(), x]
for n in range(1, 12):
    q = Q[n]
    numerator = x*q*q - 4*(q*q.derivative(2)-q.derivative()**2)
    quotient, remainder = numerator.quo_rem(Q[n-1])
    assert remainder == 0
    Q.append(quotient)
polynomials = {n: Q[n] for n in range(13)}

References

[1]

Marco Bertola and Thomas Bothner, Zeros of large degree Vorob'ev-Yablonski polynomials via a Hankel determinant identity (2014). (arXiv)

[2]

Pieter Roffelsen, Irrationality of the Roots of the Yablonskii–Vorob’ev Polynomials and Relations between Them, SIGMA 6 (2010), 095, 11 pages. (arXiv) (doi)

Links

[3]

DLMF, Section 32.8(ii): Second Painlevé equation

Similar tables

Values of the Hastings–McLeod solution of Painlevé II — gives a nonrational solution of the same differential equation at parameter zero, selected by its Airy asymptotics

Data properties

Entries are of type: integral polynomial

How well the digits are known: exact

Table is complete: no (it holds every index $0\leq n\leq12$ in this discrete family of polynomials for rational Painlevé II solutions; this initial reference segment includes every intervening index and stops at degree $78$, where the expanded entry has $1076$ characters, before $Q_{13}$ reaches $1358$ characters. The cutoff is a practical readability choice)

How they were obtained:

Computed in $\mathbb{Z}[x]$ by the defining differential recurrence, checking every division for a zero remainder. All 13 polynomials were independently reconstructed over $\mathbb{Q}[x]$ from the coefficient sum for $\exp(xt-4t^3/3)$ and the determinant in (3), using fraction-free elimination. Each coefficient is an exact integer.

The degree and leading coefficient were checked for every index; the rational Painlevé II identity was checked by clearing denominators for $1\leq n\leq12$. The defining initial values and all seven polynomials in [2], Table 1, were also compared coefficient by coefficient. The attached checks include known determinant controls and deliberate erroneous inputs. The same independent checks were repeated on the values read back from the database.