On the Ni(x) integral function and its application to the Airy's non-homogeneous equation

In this article, we discuss a recently introduced function, Ni(x), to which we will refer as the Nield-Kuznetsov function. This function is attractive in the solution of inhomogeneous Airy's equation. We derive and document some elementary properties of this function and outline its application to Airy's equation subject to initial conditions. We introduce another function, Ki(x), that arises in connection with Ni(x) when solving Airy's equation with a variable forcing function. In Appendix A, we derive a number of properties of both Ni(x) and Ki(x), their integral representation, ascending and asymptotic series representations. We develop iterative formulae for computing all derivatives of these functions, and formulae for computing the values of the derivatives at x = 0. An interesting finding is the type of differential equations Ni(x) satisfies. In particular, it poses itself as a solution to Langer's comparison equation.