  We introduce the notion of symmetric covariation, which is a new measure of
dependence between two components of a symmetric $\alpha$-stable random vector,
where the stability parameter $\alpha$ measures the heavy-tailedness of its
distribution. Unlike covariation that exists only when $\alpha\in(1,2]$,
symmetric covariation is well defined for all $\alpha\in(0,2]$. We show that
symmetric covariation can be defined using the proposed generalized fractional
derivative, which has broader usages than those involved in this work. Several
properties of symmetric covariation have been derived. These are either similar
to or more general than those of the covariance functions in the Gaussian case.
The main contribution of this framework is the representation of the
characteristic function of bivariate symmetric $\alpha$-stable distribution via
convergent series based on a sequence of symmetric covariations. This series
representation extends the one of bivariate Gaussian.
