Experimenting with Apéry Limits and WZ pairs

This article, dedicated with admiration in memory of Jon and Peter Borwein,illustrates by example, the power of experimental mathematics, so dear to them both, by experimenting with so-called Apéry limits and WZ pairs. In particular we prove a weaker form of an intriguing conjecture of Marc Chamberland and Armin Straub (in an article dedicated to Jon Borwein), and generate lots of new Apéry limits. We also rediscovered an infinite family of cubic irrationalities, that suggested very good effective irrationalitymeasures (lower than Liouville's generic 3), and that we conjectured to go down to the optimal 2. As it turned out, as pointed out by Paul Voutier (see the postscript kindly written by him), our conjectures follow from deep results in number theory. Nevertheless we believe that further experiments with our Maple programs would lead to new and interesting results.