a hard coercion for nested hyperfactorial (small gamma) functions, PART II

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One two
Three and four
Five six
Seven eight
If you get a new friend, it's a shame

Don't leave your old friend behind

  —anonymus

\(\prod_{1}^{} = \varepsilon_0\)

\(\prod_{1}^{1} = \psi(\Omega)+1\)

\(\prod_{1}^{2} = \psi(\Omega)+2\)

\(\prod_{1}^{3} = \psi(\Omega)+\omega\)

\(\prod_{1}^{\prod_{}^{3,,3}} = \psi(\Omega)+\omega^{\omega}\)

\(\prod_{1}^{\prod_{}^{3\prod_{}^{3,,3}3}} = \psi(\Omega)+\omega^{\omega}\)

\(\prod_{1}^{\prod_{1}^{}} = \psi(\Omega)+\psi(\Omega)\)

\(\prod_{1}^{\prod_{1}^{},\prod_{1}^{}} = \psi(\Omega)+\psi(\Omega)+\psi(\Omega)\)

\(\prod_{1}^{\prod_{1}^{},\prod_{1}^{}} = \psi(\Omega)\omega\)

\(\prod_{1}^{\prod_{1}^{}(\prod_{}^{3},,\prod_{}^{3})\prod_{1}^{}} = \psi(\Omega)\omega^{\omega}\)

\(\prod_{1}^{\prod_{1}^{}(\prod_{1}^{})\prod_{1}^{}} = \psi(\Omega)^2\)

\(\prod_{1}^{\prod_{1}^{}(\prod_{1}^{})\prod_{1}^{},\prod_{1}^{}(\prod_{1}^{})\prod_{1}^{},\prod_{1}^{}(\prod_{1}^{})\prod_{1}^{}} = \psi(\Omega)^2\omega\)

\(\prod_{1}^{\prod_{1}^{}(\prod_{1}^{})\prod_{1}^{}(\prod_{}^{3,,3})\prod_{1}^{}(\prod_{1}^{})\prod_{1}^{}} = \psi(\Omega)^2\omega^2\)

\(\prod_{1}^{\prod_{1}^{}(\prod_{1}^{},\prod_{1}^{})\prod_{1}^{}} = \psi(\Omega)^{\psi(\Omega)}\)

\(\prod_{1}^{\prod_{1}^{}(\prod_{1}^{}(\prod_{1}^{}()\prod_{1}^{})\prod_{1}^{})\prod_{1}^{})\prod_{1}^{}} = \psi(\Omega)^{\psi(\Omega)^{\psi(\Omega)}}\)

\(\prod_{2}^{} = \psi(\Omega+\Omega)\)

\(\prod_{2}^{\prod_{1}^{}} = \psi(\Omega+\Omega)^{\psi(\Omega)}\)

\(\prod_{2}^{\prod_{2}^{}} = \psi(\Omega+\Omega)^{\psi(\Omega+\Omega)^{}}\)

\(\prod_{3} = \psi(\Omega+\Omega+\Omega)\)

\(\prod_{3} = \psi(\Omega^2)\)

\(\prod_{\prod_{1}} = \varepsilon_{\varepsilon_{0}}\)

\(\prod_{\prod_{\prod_{1}}} = \varepsilon_{\varepsilon_{\varepsilon_{0}}}\)

\(\prod_{\prod_{\prod_{\prod_{1}}}} = \varepsilon_{\varepsilon_{\varepsilon_{\varepsilon_{0}}}}\)