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 , which has power series representation:
. 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 are such that . There is a sea of such functions, but they’re sort of slippery and hard to make