In the present paper, it is argued that non-Maxwellian distribution functions are better suited to model space plasmas. A new model distribution function called the generalized distribution function which is the generalized form of the generalized Lorentzian (kappa) distribution function has been employed to carry out theoretical investigation for parallel propagating waves in general and for Alfvén waves in particular. New plasma dispersion functions have been derived and their properties investigated. The new linear dispersion relation for Alfvén waves is investigated in detail.
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