closed form for these power series coefficients?

Question

The context for this is series approximation for perturbed iterations of the Mandelbrot set with arbitrary power $P$ (aka multibrot).

Define an arbitrary precision reference orbit: $$Z_{n+1} = Z_n^P + C$$ Using lower case for low precision differences: $$Z_{n+1} + z_{n+1} = (Z_n+z_n)^P + (C + c)$$ so: $$z_{n+1} = \sum_{p = 1}^P \begin{pmatrix} P \\ p \end{pmatrix} Z_n^{P-p} z_n^p + c$$ Now approximate the difference in iterate $z_n$ by a power series in the difference in parameter $c$: $$z_n = \sum_{k=1}^\infty A_{P,k,n} c^k$$ Combining the previous two equations gives: $$\sum_{k=1}^\infty A_{P,k,n+1} c^k = \sum_{p=1}^P \begin{pmatrix} P \\ p \end{pmatrix} Z_n^{P-p} \left(\sum_{j=1}^\infty A_{P,j,n} c^j\right)^p + c$$ Equating coefficients of $c^m$ gives a collection of iteration formulas for $A_{P,m,n+1}$ in terms of the previous coefficients and $Z_n$. These iterations are defined implicitly by the previous equation. The question is:

Is there an explicit closed form expression, perhaps in terms of nested sums, for these iteration equations for arbitrary $P$?

For example, for $P=2$ a closed form equation is: $$A_{2,1,n+1} = 2 Z_n A_{2,1,n} + 1 \\ A_{2,m,n+1} = 2 Z_n A_{2,m,n} + \sum_{k=1}^{m-1} A_{2,k,n} A_{2,m-k,n}\quad (m \gt 1)$$

Answer 1

I think the answer for arbitrary $P$ is "no". It is possible to generate the series recurrence for arbitrary fixed $P$, but the recurrence contains nested sums with depth $P-1$, which is hard to express in a closed form (that can be translated to computer code without too much effort). Some experiments with the Maxima computer algebra system:

$ maxima
(%i1) sumexpand: true$
(%i2) cauchysum: true$
(%i3) F(p, c, z) := z^p + c$
(%i4) D(p, c, z) := F(p, c + C, z + Z) - F(p, C, Z)$
(%i5) A(p) := niceindices(collectterms(expand(D(p,c,sum(a[i]*c^i,i,1,inf))),c))$
(%i6) tex(A(2))$

$$\sum_{j=2}^{\infty }{c^{j}\,\sum_{i=1}^{j-1}{a_{i}\,a_{j-i}}}+2\,Z \,\sum_{i=1}^{\infty }{c^{i}\,a_{i}}+c$$

(%i7) tex(A(3))$

$$\sum_{j=3}^{\infty }{c^{j}\,\sum_{{\it i_6}=1}^{j-2}{a_{{\it i_6}} \,\sum_{i=1}^{j-{\it i_6}-1}{a_{i}\,a_{j-{\it i_6}-i}}}}+3\,Z\, \sum_{j=2}^{\infty }{c^{j}\,\sum_{i=1}^{j-1}{a_{i}\,a_{j-i}}}+3\,Z^2 \,\sum_{i=1}^{\infty }{c^{i}\,a_{i}}+c$$

(%i8) tex(A(4))$

$$\sum_{k=4}^{\infty }{c^{k}\,\sum_{{\it i_{14}}=1}^{k-3}{a_{ {\it i_{14}}}\,\sum_{j=1}^{k-{\it i_{14}}-2}{a_{j}\,\sum_{i=1}^{k-j- {\it i_{14}}-1}{a_{i}\,a_{k-j-{\it i_{14}}-i}}}}}+4\,Z\,\sum_{j=3}^{ \infty }{c^{j}\,\sum_{{\it i_{18}}=1}^{j-2}{a_{{\it i_{18}}}\,\sum_{ i=1}^{j-{\it i_{18}}-1}{a_{i}\,a_{j-{\it i_{18}}-i}}}}+6\,Z^2\, \sum_{j=2}^{\infty }{c^{j}\,\sum_{i=1}^{j-1}{a_{i}\,a_{j-i}}}+4\,Z^3 \,\sum_{i=1}^{\infty }{c^{i}\,a_{i}}+c$$

These are not quite the desired form (have to collect a few things to get the coefficient of $c^j$) but are close enough for now. And it seems niceindices() fails... (Maxima 5.38.1 on Debian).