Integrals involving products of Airy functions, their derivatives and Bessel functions

Abstract A new integral representation of the Hankel transform type is deduced for the function F n ( x , Z ) = Z n − 1 A i ( x − Z ) A i ( x + Z ) with x ∈ R , Z > 0 and n ∈ N . This formula involves the product of Airy functions, their derivatives and Bessel functions. The presence of the latter allows one to perform various transformations with respect to Z and obtain new integral formulae of the type of the Mellin transform, K-transform, Laplace and Fourier transform. Some integrals containing Airy functions, their derivatives and Chebyshev polynomials of the first and second kind are computed explicitly. A new representation is given for the function | A i ( z ) | 2 with z ∈ C .