We wish to investigate circumstances where the solutions of (naturally given) ordinary differential equations have integer coefficients.

We start from Zagier's paper "Arithmetic and Topology of Differential Equations" and try to understand some of its features. We will then look at various examples (Apéry's recursion, hypergeometric equation,...).

Often, one observes that the solution has integral coefficients or, at least, coefficients of moderate convergence/divergence. The reason for this, however, remains a mystery.

More info available here.