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.

Additional Metadata
Keywords Riesz potentials, Airy functions, asymptotic expansions
MSC Nonlinear parabolic equations (msc 35K55)
Publisher IOP
Journal Physica Scripta
Note Proceedings of FRACTIONAL DIFFERENTIATION AND ITS APPLICATIONS (FDA 08) (ANKARA, 5–7 November 2008)
Citation
Temme, N.M, & Varlamov, V. (2009). Asymptotic expansions for Riesz potentials of Airy functions and their products. Physica Scripta, T136.