Asymptotic expansions for Riesz fractional derivatives of Airy functions and applications

Abstract

Riesz fractional derivatives of a function, $D_{x}^{\alpha}f(x)$ (also called Riesz potentials), are defined as fractional powers of the Laplacian. Asymptotic expansions for large $x$ are computed for the Riesz fractional derivatives of the Airy function of the first kind, $Ai(x)$, and the Scorer function, $Gi(x)$. Reduction formulas are provided that allow one to express Riesz potentials of products of Airy functions, $D_{x}^{\alpha}\left\{ Ai(x)Bi(x)\right\}$ and $D_{x}^{\alpha}\left\{ Ai^{2}(x)\right\}$, via $D_{x}^{\alpha}Ai(x)$ and $D_{x}^{\alpha}Gi(x)$. Here $Bi(x)$ is the Airy function of the second type. Integral representations are presented for the function $A_{2}\left(a,b;x\right)=Ai\left(x-a\right)Ai\left(x-b\right)$ with $a, b\in\mathbb{R}$ and its Hilbert transform. Combined with the above asymptotic expansions they can be used for computing asymptotics of the Hankel transform of $D_{x}^{\alpha}\left\{ A_{2}\left(a,b;x\right)\right\}$. These results are used for obtaining the weak rotation approximation for the Ostrovsky equation (asymptotics of the fundamental solution of the linearized Cauchy problem as the rotation parameter tends to zero)