Fractional derivatives of products of Airy functions

Abstract

Fractional derivatives of the products of Airy functions are investigated, $D^{\alpha }\{{\mathit{Ai}}^2(x)\}$ and $D^{\alpha }\{\mathit{Ai}(x)\times \mathit{Bi}(x)\}$, where $\mathit{Ai}(x)$ and $\mathit{Bi}(x)$ are the Airy functions of the first and second type, respectively. They turn out to be linear combinations of $D^{\alpha }\{\mathit{Ai}(x)\}$ and $D^{\alpha }\{\mathit{Gi}(x)\}$, where $\mathit{Gi}(x)$ is the Scorer function. It is also proved that the Wronskian $\mathcal{W}(x)$ of the system of half integrals $\{D^{-1/2}\mathit{Ai}(x),D^{-1/2}\mathit{Gi}(x)\}$ and its Hilbert transform $\tilde{\mathcal{W}}(x)=-H\mathcal{W}(x)$ can be considered special functions in their own right since they are expressed in terms of ${\mathit{Ai}}^2(x)$ and $\mathit{Ai}(x)\mathit{Bi}(x)$, respectively. Various integral relations are established. Integral representations for $D^{\alpha }\{\mathit{Ai}(x-a)\mathit{Ai}(x+a)\}$ and its Hilbert transform $-H{D}^{\alpha }\{\mathit{Ai}(x-a)\mathit{Ai}(x+a)\}$ are derived.