60 50 v 1 2 0 Ju n 20 06 REPRESENTATIONS AND PROPERTIES OF GENERALIZED A r STATISTICS , COHERENT STATES AND ROBERTSON UNCERTAINTY RELATIONS

The generalization of A r statistics, including bosonic and fermionic sectors, is performed by means of the so-called Jacobson generators. The corresponding Fock spaces are constructed. The Bargmann representations are also considered. For the bosonic A r statistics, two inequivalent Bargmann realizations are developed. The first (resp. second) realization induces, in a natural way, coherent states recognized as Gazeau-Klauder (resp. Klauder-Perelomov) ones. In the fermionic case, the Bargamnn realization leads to the Klauder-Perelomov coherent states. For each considered realization, the inner product of two analytic functions is defined in respect to a measure explicitly computed. The Jacobson generators are realized as differential operators. It is shown that the obtained coherent states minimize the Robertson-Schrödinger uncertainty relation.