This notebook contains additional material to the Mathematica GuideBooks  (Springer-Verlag, New York, 2004) http://www.MathematicaGuideBooks.org . It uses the stylesheet of the Mathematica GuideBooks, which is embedded.

Symbolics Volume, Chapter 1

The Yablonskii–Vorob’ev polynomials (Subscript[Q, n])(z) fulfill the following nonlinear difference-differential equation:

(Subscript[Q, n+1])(z)=((z (((Subscript[Q, n])(z))^2)-4((Subscript[Q, n])(z)(Subscript[Q, n]^″)(z)-(((Subscript[Q, n]^′)(z))^2)))/((Subscript[Q, n-1])(z))).

The initial polynomials are (Subscript[Q, 0])(z)=1 and (Subscript[Q, 1])(z)=z.

A recent article [✶] asked for a pure differential equation for the Yablonskii–Vorob’ev polynomials (Subscript[Q, n])(z). Make a polynomial ansatz for a nonlinear differential equation for the (Subscript[Q, n])(z) and determine the unknown coefficients.

Off[RuleDelayed::rhs];

Q[0][z_] := 1;
Q[1][z_] := z;
(* recursive definition *)
Q[n_][z_] := Q[n][z_] = Together[
(z Q[n - 1][z]^2 - 4 (Q[n - 1][z] Q[n - 1]''[z] - Q[n - 1]'[z]^2))/Q[n - 2][z]]

Here are the first few (Subscript[Q, n])(z).

Table[Factor @ Q[k][z], {k, 0, 5}] // TableForm

The above nonlinear difference-differential equation is of order two with respect to differentiation and is of order two with respect to differencing. So, a natural guess would be to look for a fourth order pure differential equation. We make the following ansatz:

(∑_(k,(Subscript[α, 0]),(Subscript[α, 1]),(Subscript[α, 2]),(Subscript[α, 3]),(Subscript[α, 4])=0)^d)(Subscript[c, k,(Subscript[α, 0]),(Subscript[α, 1]),(Subscript[α, 2]),(Subscript[α, 3]),(Subscript[α, 4])])(z^k)(((Subscript[Q, n])(z))^(Subscript[α, 0]))(((Subscript[Q, n]^′)(z))^(Subscript[α, 1]))(((Subscript[Q, n]^″)(z))^(Subscript[α, 2]))(((Subscript[Q, n]^((3)))(z))^(Subscript[α, 3]))(((Subscript[Q, n]^((4)))(z))^(Subscript[α, 4]))=0

Here the (Subscript[c, k,(Subscript[α, 0]),(Subscript[α, 1]),(Subscript[α, 2]),(Subscript[α, 3]),(Subscript[α, 4])])are z-independent numbers. We start with d=4 to obtain enough equations.

ansatz[Qn_, z_] := With[{d = 4, o = 4},
Sum[If[k + α0 + α1 + α2 + α3 >= o, 0, 
       c[k, α0, α1, α2, α3, α4] z^k Qn[z]^α0 * (* avoid 0^0 *)
       If[α1 == 0, 1, Qn'[z]^α1] If[α2 == 0, 1, Qn''[z]^α2]*
       If[α3 == 0, 1, Qn'''[z]^α3] If[α4 == 0, 1, Qn''''[z]^α4]],
    {k , 0, d}, {α0, 0, d}, {α1, 0, d}, 
    {α2, 0, d}, {α3, 0, d}, {α4, 0, d}]];

We get 495 equations for 290 unknowns.

Length[cls = Flatten[Table[CoefficientList[ansatz[Q[k], z], z], {k, 2, 7}]]]

Length[vars =  Union[Cases[cls, _c, Infinity]]]

We convert the equations cls into the coefficient matrix M.

M = (Plus @@ (Times @@@ Internal`DistributedTermsList[#, vars][[1]]))& /@ cls;

Next, we calculate a modular nullspace (which is faster and if a differential equation exists, it will very probably have nice small coefficients).

prime = Prime[10000];
ns = NullSpace[M, Modulus -> prime];
Length[ns]

Substituting the result in the above ansatz yields the following nonlinear fourth order differential equation.

-z (((Subscript[Q, n]^′)(z))^2)+4 (Subscript[Q, n]^((3)))(z) (Subscript[Q, n]^′)(z)-3 (((Subscript[Q, n]^(′′))(z))^2)+(Subscript[Q, n])(z) ((Subscript[Q, n]^′)(z)+z (Subscript[Q, n]^(′′))(z)-(Subscript[Q, n]^((4)))(z))=0

Factor[ansatz[𝒬[n], z] /. 
       (Rule @@@ Transpose[{vars, Mod[ns[[1]], prime, -Floor[prime/2]]}])]

We end with a quick check of the differential equation for higher order polynomials.

Table[(Qn[z] Qn'[z] - z Qn'[z]^2 + z Qn[z] Qn''[z] - 
       3 Qn''[z]^2 + 4 Qn'[z] Qn'''[z] - Qn[z] Qn''''[z] /. 
                      Qn -> Function[z, Evaluate[Q[k][z]]]) // Expand,
      {k, 30}] // Union