Barrow entropic Quintessence and Dilation dark energy Models with Generalized HDE cut-off

In the present work, we have analyzed the behaviors of extension of generalized Barrow holographic dark energy(`BHDE'). A ``generalized BHDE model based on the particle and the future horizon using infrared cut-off" was proposed by Nojiri et al. (2022). In this work, we have reviewed the generalized BHDE extension under the assumption of a generalized HDE cut-off. Using a scale factor of the form $a = k t^m$, the dynamics of the cosmos have been discussed through graphic demonstration. By applying the ``open-source emcee Python package", the values of the free parameters $k$ and $m$ are estimated on 57 OHD points by the Markov Chain Monte Carlo (MCMC) technique. We have examined the behavior of the equation of state (EoS) parameter, $( p_{de})$, and dark energy density $(\rho_{de})$. We have also discussed the equivalence of holographic dark energy (DE) with the Barrow entropic DE and its extension. Also, we have explained quintessence and dilation dark energy models in the context of Barrow entropic DE.