Riesz potentials of a function are defined as fractional powers of the Laplacian. Asymptotic expansions for $x\to\pm\infty$ are derived for the Riesz potentials of the Airy function $Ai(x)$ and the Scorer function $Gi(x)$. Reduction formulas are provided that allow to compute Riesz potentials of the products of Airy functions $Ai^2(x)$ and $Ai(x)Bi(x)$, where $Bi(x)$ is the Airy function of the second type, via the Riesz potentials of $Ai(x)$ and $Gi(x)$. Integral representations are given for the function $A_{2}\left(a,b;x\right)=Ai\left(x-a\right)Ai\left(x-b\right)$ with $a, b\in \mathbf{R}$, and its Hilbert transform. Combined with the above asymptotic expansions they can be used for obtaining asymptotics of the Hankel transform of Riesz potentials of $A_{2}(a,b;x)$. The study of the above Riesz fractional derivatives can be used for establishing new properties of Korteweg-de Vries-type equations.