It is known that if $X$, $X_1$ and $X_2$ are iid random draw from a continuous stable distribution, then $X_1 + X_2$ has the same distribution as $a + b X$. Parameters $a$ and $b$ are determined by $$E[X_1 + X_2] = E[a + b X],$$ $$V [X_1 + X_2] = V[a + b X],$$ where $V$ represents variance.
Is there a similar way of determining the distribution of $X_1+X_2$ in terms of $X$ (say, $a + b X$ if that is possible) if our three random variables are iid draws from a discrete stable distribution?