In 1978 Roger Apéry proved irrationality of zeta(3) approximating it by ratios of terms of two sequences of rational numbers both satisfying the same recurrence relation. His study of the growth of denominators in these sequences involved complicated explicit formulas for both via sums of binomial coefficients. Subsequently, Frits Beukers gave a more enlightening proof of their properties, in which zeta(3) can be seen as an entry in a monodromy matrix for a differential equation arising from a one-parametric family of K3 surfaces. In the talk I will define Apéry constants for Fuchsian differential operators and explain the generalized Frobenius method due to Golyshev and Zagier which produces an infinite sequence of Apéry constants starting from a single differential equation. I will then show a surprising property of their generating function and conclude that the Apéry constants for a geometric differential operator are periods.
This is work in progress with Spencer Bloch and Francis Brown.[-]