User blog comment:Moooosey/Simple psi-- a fairly simple ocf/@comment-43798125-20191105221608/@comment-35470197-20191106010629

With respect to the updated definition, \(\psi_0(\alpha) = \varphi(1,0,\alpha)\) for any \(\alpha < \Omega\), \(\psi_0(\Omega) = \varphi(1,1,0)\), \(\psi_0(\Omega^{\omega}) = \varphi(1,0,\ldots,0)\), and so on. It is well-known as a "Veblen-based OCF", and its limit coincide with \(\psi_{\Omega}(\Omega_2)\) with respect to the standard OCF based on weakly Mahlo cardinal.

By the way, since \(\psi_1(\alpha)\) is just the constant \(\Gamma_{\Omega+1}\), I guess that you made another typo in the definition of \(\psi_1\).