$$t^2 \frac{\partial^3}{\partial t^3}\Delta_t(s)+s^2 \frac{\partial}{\partial s} \Delta_t(s)=0 $$
is satisfied by
$$\Delta_t(s)= - d(s) \sqrt{\frac{t}{s}}Y_1{(4\pi\sqrt{ts})}- d(s)\sqrt{\frac{t}{s}}K_1(4\pi \sqrt{ts})$$
for $Y_1$ and $K_1$ Bessel functions and $d(s)$ is the divisor function.
$$\Delta(t)= -\sum_{s \in \Bbb N} d(s) \sqrt{\frac{t}{s}}Y_1{(4\pi\sqrt{ts})}-\sum_{s \in \Bbb N}^\infty d(s)\sqrt{\frac{t}{s}}K_1(4\pi \sqrt{ts})$$
Let $A$ be set such that $n(A)=5$. How many functions can we define on $A$ with the property $(f\circ f)(x)=x$ ?
I think the identity function works but what about others? Should $f$ have an inverse? I think permutations may be involved, but I am not sure how to progress.
