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We derive the first few terms of the asymptotic expansion of the Fredholmequations for one-dimensional $\kappa$-component fermions with repulsive andwith attractive delta-function interaction in strong and weak coupling regimes.We thus obtain a highly accurate result for the ground state energy of amulticomponent Fermi gas with polarization for these regimes. This resultprovides a unified description of the ground state properties of the Fermi gaswith higher spin symmetries. However, in contrast to the two-component Fermigas, there does not exist a mapping that can unify the two sets of Fredholmequations as the interacting strength vanishes. Moreover, we find that thelocal pair correlation functions shed light on the quantum statistic effects ofthe $\kappa$-component interacting fermions. For the balanced spin case withrepulsive interaction, the ground state energy obtained confirms Yang and You'sresult [Chin. Phys. Lett. {\bf 28}, 020503 (2011)] that the energy per particleas $\kappa \to \infty$ is the same as for spinless Bosons.

1D multicomponent Fermions with delta function interaction in strong and weak coupling limits: $\kappa$-component Fermi gas