# “ethereal” functions

Say a function is ethereal iff:
1. all the coefficients in the power series representation of it are transcendental
2. there is no easy way of getting at those coefficients

2 is a crapshoot. Let me give a somewhat contrived example, take the Airy function $Bi(z)$, which has power series representation:
$\mathrm{Bi}(z) = \frac{1}{\pi\sqrt[6]{3}} \sum_{n=0}^{\infty} \frac{3^{n/3} \Gamma((1+n)/3) \sin(2(1+n)\pi/3)}{k!}$. I don’t consider this function ethereal because the coefficients are easy to get at, so to speak.

Now consider a function whose power series coefficients $a_{n}$ are such that $\mathrm{Bi}(a_{n})=n \forall n\in\mathbb{N}$. There is a sea of such functions, but they’re sort of slippery and hard to make