  Bifractional transformations which lead to quantities that interpolate
between other known quantities, are considered. They do not form a group, and
groupoids are used to described their mathematical structure. Bifractional
coherent states and bifractional Wigner functions are also defined. The
properties of the bifractional coherent states are studied. The bifractional
Wigner functions are used in generalizations of the Moyal star formalism. A
generalized Berezin formalism in this context, is also studied.
