So, the bonus points of all this.......
\[
\vartheta(x,z) = \sum_{n=0}^\infty f_\theta^{\circ q n}(z) \frac{(-x)^n}{n!}\\
\]
Then:
\[
\frac{d^s}{dx^s}\Big{|}_{x=0} \vartheta(x,z) = \lambda^{qs}z + O(z^2)\\
\]
The value \(\lambda^{s+as} = e^{i\theta s}\), for some \(a\). Then:
\[
\frac{d^{\frac{s+as}{q}}}{dx^{\frac{s+as}{q}}}\Big{|}_{x=0} \vartheta(x,z) = e^{i\theta s}z + O(z^2)\\
\]
The differintegral above, always satisfies a semigroup operation:
\[
D^{s_0} D^{s_1} = D^{s_0 + s_1}\\
\]
And this is carried over in the Mellin transform, from that, it is just like the exponential function
It's just like the semi-groups of Gottfried's matrices; but it's the integral interpretation.
But as \(\mathbb{H}^* \to \mathbb{H}\), we have that \(a \to \infty\). To ensure convergence, we need to bound the growth of \(a\). Bounding the growth of \(a\) is a deep problem in analytic number theory... It has everything to do with the density of the solutions \( j \equiv p \pmod{q}\)....
This will absolutely produce an integral though.... as \( a \to \infty\), we can have an integral interchange. It's escaping me at the moment. But we're going to get something really nice.
EDIT:
It's also important to note that:
\[
f_\theta^{\circ s}(z)\\
\]
Will be holomorphic when \(|e^{i\theta s}| < 1\), where by \(|z| < \delta\) is fixed for \(\delta>0\). So long as \(\theta \in \mathbb{H}^*\)--by which the set \(\mathbb{H}\), can at worst be a boundary behaviour \(|e^{i\theta s} | \le 1\) when \(|z| \le \delta\). This is enough to give us some stability to this construction. But getting everything to work, we need to talk about \(j \equiv p \pmod{q}\)... We don't need the super advanced shit. Just some old timey guesstimates....
\[
\vartheta(x,z) = \sum_{n=0}^\infty f_\theta^{\circ q n}(z) \frac{(-x)^n}{n!}\\
\]
Then:
\[
\frac{d^s}{dx^s}\Big{|}_{x=0} \vartheta(x,z) = \lambda^{qs}z + O(z^2)\\
\]
The value \(\lambda^{s+as} = e^{i\theta s}\), for some \(a\). Then:
\[
\frac{d^{\frac{s+as}{q}}}{dx^{\frac{s+as}{q}}}\Big{|}_{x=0} \vartheta(x,z) = e^{i\theta s}z + O(z^2)\\
\]
The differintegral above, always satisfies a semigroup operation:
\[
D^{s_0} D^{s_1} = D^{s_0 + s_1}\\
\]
And this is carried over in the Mellin transform, from that, it is just like the exponential function
It's just like the semi-groups of Gottfried's matrices; but it's the integral interpretation.But as \(\mathbb{H}^* \to \mathbb{H}\), we have that \(a \to \infty\). To ensure convergence, we need to bound the growth of \(a\). Bounding the growth of \(a\) is a deep problem in analytic number theory... It has everything to do with the density of the solutions \( j \equiv p \pmod{q}\)....
This will absolutely produce an integral though.... as \( a \to \infty\), we can have an integral interchange. It's escaping me at the moment. But we're going to get something really nice.
EDIT:
It's also important to note that:
\[
f_\theta^{\circ s}(z)\\
\]
Will be holomorphic when \(|e^{i\theta s}| < 1\), where by \(|z| < \delta\) is fixed for \(\delta>0\). So long as \(\theta \in \mathbb{H}^*\)--by which the set \(\mathbb{H}\), can at worst be a boundary behaviour \(|e^{i\theta s} | \le 1\) when \(|z| \le \delta\). This is enough to give us some stability to this construction. But getting everything to work, we need to talk about \(j \equiv p \pmod{q}\)... We don't need the super advanced shit. Just some old timey guesstimates....