A very rich interplay between arithmetic, geometry, transcendence and combinatorics arises in the study of homogeneous linear differential equations and especially of those that “come from geometry” and the related study of Siegel G-functions. A remarkable result is that, by adding variables, we can see many transcendental G-functions (and thus many generating series) as arising in a natural way from much more elementary function, namely rational functions. This process, called diagonalization, can be thought of as a formal integration. I will discuss some properties enjoy by diagonals of rational functions and connect them with Lucas'congruences for binomial coefficients and algebraic independence of power series. This corresponds to some joint works with Jason Bell and Eric Delaygue.