The Hyperbolic distribution is characterized by a single log change of coordinates on the PDF yielding a hyperbola.
What if we impose a double log change of coordinates on the PDF yielding a hyperbola? The PDF $\varphi_s(x)$ satisfies this constraint.
And my question:
Does the probability distribution function$$ \varphi_s(x)=\frac{e^{\frac{s}{\log(x)}}}{2\sqrt{s}K_1(2\sqrt{s})} $$
already appear in some context or has a name or is part of a wider class of distributions?
I noticed two curious things... $K_1$ (modified Bessel function of second kind) also appears in the PDF for the Hyperbolic distribution, which is characterized by the logarithm of the PDF being a hyperbola (a log change of coordinates).
Another interesting part is that a log-log change of coordinates actually makes $\varphi_s(x)$ into a hyperbola. This can be verified by the computation $\log\big(\varphi_s(e^x)\big).$ So you could in a similar vein, state that a double log change of coordinates being a hyperbola characterizes the PDF $\varphi_s(x).$
Maybe $\varphi_s(x)$ is a random mixture of some unknown distributions? (The Hyperbolic distribution is a random mixing of normal distributions).