A Tutorial on Closed Difference Forms

Closed difference forms allow to prove many ($q$-)hypergeometric (multi-)sum identities. As a spin-off, these proofs lead to the discovery of ``companion'' identities. To find more companion identites, we transform closed difference forms in a nontrivial way that preserves their closedness, similarly to a change of variables (pullback) of differential forms. New identities found by this method will be presented.