Yes, Tommy!
That's very much my point. This finds the complimentary solution. I think we can work a tad differently though; so let me take your example but flip it slightly on its head. Let:
\[
f(z) = 5z + z^3\\
\]
Now define the function:
\[
\gamma_c(w) = \Omega_{j=1}^\infty \frac{w f(z)}{w+c^j}\,\bullet z\\
\]
For \((|c| > 1\). This function is holomorphic in \(c\) and \(w\), so long as \(w \neq -c^j\) for \(j \ge 1\). And further, is a meromorphic function \(\gamma_c(w) : \mathbb{C}^2 \to \widehat{C}\). This function satisfies:
\[
\gamma_c(cw) = \frac{w \gamma_c(w)}{w+1}\\
\]
Now, we can take the iteration:
\[
\Psi_c(w) = \lim_{k\to\infty} f^{-k} \gamma(c^k w)\\
\]
This iteration only converges for \(|c| > 5\).
My point is that, if we look at, instead, your summation condition, (like the one you wrote), we should be able to solve if for \(1 < |c| < 5\)...
The uniqueness of these equations is difficult to describe--and it requires understanding the uniqueness of these almost elliptic functions \(\theta(z+1) = \theta(z)\) and \(\theta(z+2 \pi i/\log( c)) = \theta(z) - 2\pi i/\log(c )\). These functions are unique UPTO an elliptic function and the location of the singularities and their singular parts.
Suppose:
\[
\theta_1 \neq \theta_2\\
\]
Suppose every where \(\theta_1\) has a singularity \(z_0\), so does \(\theta_2\). Suppose additionally, the singular parts of this singularity are equivalent (this is difficult to describe properly, so I'll use a weaker condition at the moment):
\[
\lim_{z \to z_0}|\theta_1(z) - \theta_2(z)| < \infty\\
\]
Then, \(\theta_1 = \theta_2 + C\) for a constant \(C\), because:
\[
\theta_1(z) - \theta_2(z) = \wp(z)
\]
And:
\[
\wp(z+1) = \wp(z)\,\,\,\,\wp(z+2\pi i/\log( c)) = \wp(z)\\
\]
The function \(\wp(z)\) has no singularities. Therefore it is bounded on \(\mathbb{C}\). By Liouville's theorem, therefore it is constant.
So essentially, any solution to these types of equations, only differ by the singular behaviour of their theta function... Which equates to the statement that they are equivalent upto an elliptic function!
That's very much my point. This finds the complimentary solution. I think we can work a tad differently though; so let me take your example but flip it slightly on its head. Let:
\[
f(z) = 5z + z^3\\
\]
Now define the function:
\[
\gamma_c(w) = \Omega_{j=1}^\infty \frac{w f(z)}{w+c^j}\,\bullet z\\
\]
For \((|c| > 1\). This function is holomorphic in \(c\) and \(w\), so long as \(w \neq -c^j\) for \(j \ge 1\). And further, is a meromorphic function \(\gamma_c(w) : \mathbb{C}^2 \to \widehat{C}\). This function satisfies:
\[
\gamma_c(cw) = \frac{w \gamma_c(w)}{w+1}\\
\]
Now, we can take the iteration:
\[
\Psi_c(w) = \lim_{k\to\infty} f^{-k} \gamma(c^k w)\\
\]
This iteration only converges for \(|c| > 5\).
My point is that, if we look at, instead, your summation condition, (like the one you wrote), we should be able to solve if for \(1 < |c| < 5\)...
The uniqueness of these equations is difficult to describe--and it requires understanding the uniqueness of these almost elliptic functions \(\theta(z+1) = \theta(z)\) and \(\theta(z+2 \pi i/\log( c)) = \theta(z) - 2\pi i/\log(c )\). These functions are unique UPTO an elliptic function and the location of the singularities and their singular parts.
Suppose:
\[
\theta_1 \neq \theta_2\\
\]
Suppose every where \(\theta_1\) has a singularity \(z_0\), so does \(\theta_2\). Suppose additionally, the singular parts of this singularity are equivalent (this is difficult to describe properly, so I'll use a weaker condition at the moment):
\[
\lim_{z \to z_0}|\theta_1(z) - \theta_2(z)| < \infty\\
\]
Then, \(\theta_1 = \theta_2 + C\) for a constant \(C\), because:
\[
\theta_1(z) - \theta_2(z) = \wp(z)
\]
And:
\[
\wp(z+1) = \wp(z)\,\,\,\,\wp(z+2\pi i/\log( c)) = \wp(z)\\
\]
The function \(\wp(z)\) has no singularities. Therefore it is bounded on \(\mathbb{C}\). By Liouville's theorem, therefore it is constant.
So essentially, any solution to these types of equations, only differ by the singular behaviour of their theta function... Which equates to the statement that they are equivalent upto an elliptic function!