Airy functions and Riesz fractional trigonometric operators

In this paper, using the Riesz fractional derivative D:=−d2/dx2, we present the Mellin transforms of shifted Airy functions in terms of the Riesz fractional derivatives of Airy functions. We also show some integral addition formulas for the products of Airy functions in terms of the Riesz fractional trigonometric operators. Moreover, the fundamental solution of the Ostrovsky equation is given by the Riesz fractional trigonometric operators.