Apologies for the self promotion---the preprint "A q-analog of Ljunggren's binomial congruence", arxiv.org/abs/1103.3258, establishes a q-analog of the binomial congruence modulo $p^3$.

Thank you, Piero! I finally got hold of a copy of Hörmander's book and that's exactly what I was looking for.

Very nice and clear explanation! Do you know of a reference (for citing) which contains this?

Because it's for a paper where this is a technical detail: a function f(x) which is not known to be smooth (and in fact isn't everywhere) has a Mellin transform which satisfies a functional equation. By the distributional Mellin formulation it follows that f(x) is the weak solution of a differential equation. To conclude that f(x) is analytic on certain intervals, we currently appeal to elliptic regularity. While that certainly works (and is just one sentence plus reference) it feels a bit overpowered for the case of an ODE.

Thanks for your answer! That only asserts the existence of real analytic solutions, doesn't it? Together with classical uniqueness, one still needs a result which says that weak solutions (in our case) are indeed classical. Do you know a nice reference for that?

Thank you! Regularity via uniqueness is an interesting point. I'm not sure about the initial conditions though. It seems that getting derivative values at one point from a distributional solution (which we just know to be a solution; no initial conditions) requires the smoothness (well, at least some smoothness) that we are trying to establish.

Probably I didn't phrase the question well enough: For an audience which may not be familiar with the theory of elliptic operators, what would be the proper way to (in one sentence plus a reference) justify that a weak solution in the case at hand is automatically real analytic? Currently, I have to make reference to a textbook on PDEs which treats elliptic regularity even though the problem for linear ODEs seems so much simpler. I very much appreciate your nice outline though!