Derivatives, Integrals and Polynomials Arising from the Inhomogeneous Airy’s Equation

The various forms of Airy’s differential equation are discussed in this work together with the special functions that arise in the processes of their solutions. Further properties of the arising integral functions are introduced and their connections to existing special functions are derived. A generalized form of the Scorer functions is obtained and expressed in terms of the generalized Nield-Kuznetsov functions. Complementary functions to the Nield-Kuznetsov functions are obtained together with higher derivatives of all generalized functions arising in this work. Airy’s polynomials and generalized Airy’s polynomials are derived and defined iteratively.