  The classic central limit theorem and $\alpha$-stable distributions play a
key role in probability theory, and also in Boltzmann-Gibbs (BG) statistical
mechanics. They both concern the paradigmatic case of probabilistic
independence of the random variables that are being summed. A generalization of
the BG theory, usually referred to as nonextensive statistical mechanics and
characterized by the index $q$ ($q=1$ recovers the BG theory), introduces
special (long range) correlations between the random variables, and recovers
independence for $q=1$. Recently, a $q$-central limit theorem consistent with
nonextensive statistical mechanics was established
\cite{UmarovTsallisSteinberg} which generalizes the classic Central Limit
Theorem. In the present paper we introduce and study symmetric
$(q,\alpha)$-stable distributions. The case $q=1$ recovers the L\'evy
$\alpha$-stable distributions.
