Hypergeometry inspired by irrationality questions

Diophantine approximations are normally designed for proving the irrationality and linear independence
of numbers.
In my talk, I am interested in a different aspect of rational diophantine approximations to concrete mathematical quantities (namely, to Catalan's constant, $\pi^2$ and $\log2$): the approximations are used as a source of (quite non-trivial!) hypergeometric identities.
The talk is based on joint work in progress with Christian Krattenthaler (Vienna).