Fractional derivatives of products of Airy functions

Fractional derivatives of the products of Airy functions are investigated, Dα{Ai 2(x)} 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 WronskianW(x) of the system of half integrals {D−1/2Ai(x),D−1/2Gi(x)} and its Hilbert transform W(x)=−HW(x) can be considered special functions in their own right since they are expressed in terms of Ai 2(x) 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. © 2007 Elsevier Inc. All rights reserved