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 Adler–Moser polynomials (Subscript[θ, k])(z) [✶] defined recursively through

(Subscript[θ, n]^′)(z)(Subscript[θ, n-2])(z)- (Subscript[θ, n-2]^′)(z)(Subscript[θ, n])(z)=(2 n-1) (((Subscript[θ, n-1])(z) )^2)

with the initial conditions (Subscript[θ, 0])(z)=1, (Subscript[θ, 1])(z)=z+(Subscript[c, 1]) depend on n free parameters (Subscript[c, k]), 1≤k≤n. 
The integrals ∫((((Subscript[θ, n])(z)/(Subscript[θ, n-1])(z)))^2)dz and ∫((((Subscript[θ, n-1])(z)/(Subscript[θ, n])(z)))^2)dz are rational functions and contain no logarithms [✶]. Verify this statement for the first few n.
For any generic complex values of the (Subscript[c, k]), the roots of (Subscript[θ, n-1])(z) and (Subscript[θ, n])(z) are the equilibrium positions of n(n-1)/2 positive and (n+1)(n+2)/2 negative unit charges that attract each other through a logarithmic potential in the plane [✶], [✶]. For some parametrized (Subscript[c, k]), show some such equilibrium configurations.

Off[RuleDelayed::rhs];
θ[0][z_] := 1;
θ[1][z_] := z + c[1];
θ[n_][z_] := θ[n][z_] = 
θ[n - 2][z] (c[n] + Integrate[(2 n θ[n - 1][z]^2 - θ[n - 1][z]^2)/θ[n - 2][z]^2, 
                              z]) // Together // Cancel

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

θ[2][z]

θ[3][z]

θ[4][z]

The (Subscript[θ, n])(z) grow quickly in size.

θ[5][z] // Expand // Length

Replacing the (Subscript[c, k]) by (λ^(2k-1))(Subscript[c, k]) and z by λ z yields a homogeneous polynomial of order (λ^(n(n+1)/2)) [✶] .

Table[(Factor[θ[n][z] /. {z -> λ z, c[k_] :> λ^(2k - 1) c[k]}]) /.
      _Plus :> 1, {n, 0, 5}]

Despite the parameters (Subscript[c, k]) present in the (Subscript[θ, n])(z), the above given integrals are rational functions in z. Here are the first few of these integrals explicitly calculated.

Table[{n, Integrate[(θ[n - 1][z]/θ[n][z])^2, z]}, {n, 1, 3}]

Table[{n, Integrate[(θ[n][z]/θ[n - 1][z])^2, z]}, {n, 1, 3}]

For visualizing the position of the zeros of consecutive (Subscript[θ, n])(z), we implement the function chargeGraphics.

chargeGraphics[polyPair_, z_, {φ_, pp_}] := 
Module[{r1, r2}, 
Graphics[{PointSize[0.004], 
Table[{r1, r2} = NRoots[# == 0, z]& /@ polyPair;
{{Hue[0.0], Point[{Re[#], Im[#]}]& /@ Cases[r1, _?NumberQ, Infinity]},
 {Hue[0.8], Point[{Re[#], Im[#]}]& /@ Cases[r2, _?NumberQ, Infinity]}}, 
     {φ, 0, 2Pi, 2Pi/pp}]}, AspectRatio -> Automatic]];

In the following two graphics, the (Subscript[c, k]) are parametrized concentric circles and Lissajou figures.

Show[GraphicsArray[
{(* c[k] moves on concentric circles *)
 chargeGraphics[{θ[4][z], θ[5][z]} /. {c[k_] :> k Exp[I φ]}, z, {φ, 480}],
 (* c[k] moves on Lissajou figures *)
 chargeGraphics[{θ[4][z], θ[5][z]} /. 
              {c[1] :> Cos[φ] + I Sin[φ], c[2] :> Cos[2 φ] + I Sin[3 φ],
               c[3] :> Cos[5 φ] + I Sin[1 φ], c[4] :> Cos[2 φ] + I Sin[5 φ],
               c[5] :> Cos[1 φ] + I Sin[3 φ]}, z, {φ, 240}]}]]

The next two graphics use random smooth functions for the (Subscript[c, k]).

(* random, smooth, periodic function *)
𝕣 := Sum[Random[] Exp[I k (φ + 2Pi Random[])], {k, 0, 3}]

SeedRandom[111];
Show[GraphicsArray[Table[
chargeGraphics[{θ[4][z], θ[5][z]} /. 
               Table[c[k] -> 𝕣, {k, 5}], z, {φ, 600}], {2}]]]