the following equation is from "[1970] Goodson - Distributed system simulation using infinite product expansions":
\begin{align*} \cosh z + \left( c z+ \frac{d}{z} \right) \sinh z & = (1 + d) \displaystyle\prod_{n=1}^{\infty} \left( 1 + \frac{z^2}{p_n^2} \right)\\ \tan p_n & = \frac{p_n}{c p_n^2 - d}, \quad p_n \ge 0, \text{real} \end{align*}
I am not sure if $p_n \ge 0$ is correct, I guess it is $p_n >0$ instead. Maybe someone can give a reference (e.g. a book) where to find the derivation of this equation (or just the equation itself)? This would help me a lot!
Best
The expansion is a nice example of the Weierstrass factorization theorem.
Two well known factorization examples are
$$ \sinh(z) = z \prod_{n=1}^\infty \left( 1 + \frac{z^2}{(\pi\, n)^2}\right), \quad \cosh(z) = \prod_{n=1}^\infty \left( 1 + \frac{z^2}{(\pi\, n -\pi/2)^2}\right). $$
Define functions
$$ w = g(z) := \coth^{-1}( c z + d/z) $$
and
$$ f(z) := \cosh(z) + (c z + d/z) \sinh(z). $$
Now
$$ f(z) = \cosh(z) + \coth(w) \sinh(z)= \sinh(z+w)/\sinh(w). $$
As a power series
$$ f(z) = (1 + d) + (3 + 6 c + d) z^2/3! + (5 + 20 c + d) z^4/5! + O(z^6). $$
Get the zeros of $\, f(z)\,$ for its infinite product expansion as follows. Let $\, p_n \approx (n-1)\, \pi \,$ with $\,n>1\,$ be the $n$-th root of $\, f(i\,t). \,$ That is, $\, f(i\, p_n) = 0. \,$
Note that the first root $\, p_1 \approx \pi/2. \,$ The Weierstrass factorization theorem implies that
$$\, f(z) = (1 + d) \prod_{n=1}^\infty \left( 1 + \frac{z^2}{p_n^2}\right) $$
with decent convergence. Since $\, f(i\, u) = \sinh(i\, u + w)/\sinh(w) = 0 \,$ iff $\, w = - i\, u + i\, n \pi, \,$ make the substitution $\, z = i\, t, \,$ to get
$$ \tan(u) = i \tanh(w) = 1 / (c\, z + d / z) = 1 / (c\, t - d / t) = t / (c\, t^2 - d). $$
Now let $\, u = p_n\,$ to finally get
$$ \tan(p_n) = p_n / (c\, p_n^2 - d). $$