In this work, we derive a dispersion equation that describes the excitation
of the oblique (or Alfv\'en) firehose instability in a plasma that contains
both electron and ion species modelled by bi-kappa velocity distribution
functions. The equation is obtained with the assumptions of low-frequency waves
and moderate to large values of the parallel (respective to the ambient
magnetic field) plasma beta parameter, but it is valid for any direction of
propagation and for any value of the particle gyroradius (or Larmor radius).
Considering values for the physical parameters typical to those found in the
solar wind, some solutions of the dispersion equation, corresponding to the
unstable mode, are presented. In order to implement the dispersion solver,
several new mathematical properties of the special functions occurring in a
kappa plasma are derived and included. The results presented here suggest that
the superthermal characteristic of the distribution functions leads to
reductions to both the maximum growth rate of the instability and of the
spectral range of its occurrence
