I was reading the Patterson's paper "The limit set of a fuchsian group" and I came across something intriguing. Given a fuchsian group $\Gamma$, a certain measure was constructed by means of the series:
$$f(z, w, s) = \sum_{\gamma \in \Gamma}h(z, \gamma(w))^{-s}$$
Where:
$$h(z, w) = \frac{|1 - z^*w|^2}{(1 - |z|^2)(1 - |w|^2)}$$
And the exponent of convergence $\delta(\Gamma)$ of this series is defined as the infimum of the exponents such that $f(z, w, s)$ converges so that it diverges if $s < \delta(\Gamma)$ and converges if $s > \delta(\Gamma)$.
Patterson cited some papers of Beardon [1, 2] which presents some estimates upon the exponent of convergence. However, in those papers, Beardon talks about the Poincáre series as usually defined in books, i. e.:
$$\theta(z) = \sum_{\gamma \in \Gamma}|\gamma'(z)|^{-s}H(\gamma(z))$$
Where $H$ is a rational function.
One of my questions is how to proof that the exponent of convergence of both $f(z, w, s)$ and $\theta(z)$ are the same.
Moreover, practically all other sources on the subject uses a third type of series, what they call Poicaré series but is:
$$P(z, w) = \sum_{\gamma \in \Gamma}\exp(-sd(z, \gamma(w))$$
Where $d$ is the hyperbolic distance.
Is the exponent of convergence of $P$ the same of the other series? Some clues of how to prove it?
At last, if anyone could give some intuition of how the series are related to each other or the intuition behind them, I would be grateful.
[1] The exponent of convergence of Poincaré series - A. F. Beardon
[2] Inequalities for certain Fuchsian groups - A. F. Beardon