Vieta Formula, Distributions, and Lemniscate

We give two extensions of the classical Vieta (or Viète) formula.
The first extension leads to the classical Fabius function, an infinitely differentiable function that is nowhere analytic. The second extension discusses the corresponding formula for
the elliptic curves with complex multiplication $y^2=x^3 -Dx$, $y^2=x^3-D$.