双曲線関数の逆数和の連分数展開
https://mathlog.info/articles/3408 より、
\begin{eqnarray}
\sum_{n=0}^{\infty} \frac{z^{n}}{a r^{n}-b q^{n}} &=&
\kfrac_{n=0}^{\infty} \frac{v_{n}}{a r^{n}-b q^{n}} \\
\end{eqnarray}
\begin{eqnarray}
\left\{\begin{array}{l}
v_{0} &=&1 \\
v_{2n-1} &=&-(rq)^{n-1}\left(a r^{n-1}-b q^{n-1}\right)^{2} z \\
v_{2n} &=&-ab(rq)^{n-1}\left(r^{n}-q^{n}\right)^{2} z
\end{array}\right.
\end{eqnarray}
$\cosh_{\pm}x := \large\frac{e^x \pm e^{-x}}{2}$
\begin{eqnarray}
\sum_{n=0}^{\infty} \frac{z^{n}}{\cosh_{\pm}(an+b)} &=&
2\sum_{n=0}^{\infty}
\frac{z^n}{e^{an+b} \pm e^{-an-b}} \\&=&
\kfrac_{n=0}^{\infty} \frac{v_{n}}{\cosh_{\pm}(an+b)} \\
\end{eqnarray}
\begin{eqnarray}
\left\{\begin{array}{l}
v_{0} &=& z \\
v_{2n-1} &=& -\cosh_{\pm}^2(a(n-1)+b) z \\
v_{2n} &=& \pm\sinh^2(an) z
\end{array}\right.
\end{eqnarray}