Fractional derivatives of products of Airy functions

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