Direct and Transform Methods to Higher Derivatives of Ki(x)

Abstract

Higher derivatives and associated polynomials of the standard Nield-Kuznetsov function of the second kind are investigated in this work. Two approaches are introduced in this work. The first, is the direct method of differentiation and generalization of the nth derivative. This approach is dependent on higher derivatives of the Nield-Kuznetsov function of the first kind. The second is the transform method in which integral transforms associated with the Nield-Kuznetsov function of the second kind are introduce first, and higher derivatives are then obtained. The transform method is independent of the direct higher derivatives of the Nield-Kuznetsov function of the first kind. Both approaches are important in practical and theoretical mathematical analysis, and both give rise to associated Airy polynomials, discussed in this work.