  We generalize Hagedorn's statistical theory of momentum spectra of particles
produced in high-energy collisions using Tsallis' formalism of non-extensive
statistical mechanics. Suitable non-extensive grand canonical partition
functions are introduced for both fermions and bosons. Average occupation
numbers and moments of transverse momenta are evaluated in an analytic way. We
analyse the energy dependence of the non-extensitivity parameter q as well as
the q-dependence of the Hagedorn temperature. We also take into account the
multiplicity. As a final result we obtain formulas for differential cross
sections that are in very good agreement with e+e- annihilation experiments.
