Derivatives, Integrals, and Polynomials Arising from the Inhomogeneous Airy 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 discussed, and their connections to existing special functions are derived. A generalized form of the Scorer function is obtained and expressed in terms of the generalized Airy and Nield–Kuznetsov functions. Higher derivatives of all generalized functions arising in this work are obtained together with their associated generalized Airy polynomials. A computational procedure for the generalized Scorer function is introduced and applied to computing and graphing it for different values of its index. The solution of an initial value problem involving the generalized Scorer function is obtained.