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

A new integral representation of the Hankel transform type is deduced for the function F(n)(x, Z) = Z(n-1) Ai(x - Z)Ai(x + Z) with x is an element of R, Z > 0 and n is an element of 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 |Ai(z)|(2) with z is an element of C. (C) 2010 Elsevier Inc. All rights reserved.