Generalized dispersion relations for Alfvén waves are derived, accounting for arbitrary values of ω/ωci, k⊥ρi, and k⊥λe, where ω(ωci) is the wave (ion gyro) frequency, k⊥ = 2π/λ⊥ the perpendicular (to the external magnetic field line) wavenumber, and ρi(λe) the ion gyroradius (electron skin depth). Our dispersion relations are appropriate for deducing the relationships of the dispersive (inertial and kinetic) Alfvén waves with other plasma modes. The present results can be considered as a prerequisite for understanding some salient features of the broadband electromagnetic/electrostatic waves that are frequently observed by the Freja and FAST spacecraft in the auroral zone of the Earth's ionosphere.