User blog comment:Alemagno12/A ordinal notation/@comment-11227630-20150721045807

$$A(0,0)=\omega,\ A(0,\alpha+1)=A(0,\alpha)+\omega$$, so $$A(0,\alpha)=\omega(1+\alpha)$$. For $$A(\alpha+1,0)$$ and $$A(\alpha+1,\beta+1)$$, the rules are the same as $$\varphi$$ function. As a result, $$A(\Omega,\alpha)=\Gamma_0$$.

However, you don't define $$A(\alpha,\beta)$$ for limit ordinal $$\alpha$$ or $$\beta$$, so they fail, from $$A(0,\omega)$$ on.