Homogeneous Finsler spaces with some special $(\alpha ,\beta )$-metrics

In this paper, first we prove the existence of invariant vector field on a homogeneous Finsler space with infinite series $(\alpha ,\beta )$-metric and exponential metric. Next, we deduce an explicit formula for the the $S$-curvature of homogeneous Finsler space with these metrics. Using this formula, we further derive the formula for mean Berwald curvature of the homogeneous Finsler space with the above mentioned metrics.