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

摘要

A new integral representation of the Hankel transform type is deduced for the function F_n(x,Z)=Zn-1Ai(x-Z)Ai(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 |Ai(z)|2 with z∈C.