On the characteristic function of the joint limit distribution of the first and second order power sums

In this paper we investigate the joint limit characteristic function of normalized first and second-order power sums for a sequence of i.i.d. random variables with distributions belonging to the domain of attraction of an $\alpha$-stable law for $\alpha< 2$. An explicit form is presented in terms of generalized hypergeometric functions when $0 < \alpha < 2$, $\alpha\ne 1$, and that of Fresnel functions $S$ and $C$ for $\alpha=1$.