While discussing the small oscillations of particles about a stable equilibrium, Landau writes

Where q is the generalized co-ordinate.

Section 21, Volume 1

1. How do you know such a polynomial expansion for q is allowed? How do you know it exists? After all, this is any old U with its first derivative 0 at q_{0}.
2. Why does the co-efficient of [tex]\dot{q}^{2}[/tex] have to be a function of q? I thought it'd be a constant.

You can just think of this formula as something that applies for any potential which is twice differentiable at q_{0}. Most physically meaningful potentials will of course be infinitely differentiable.