Connections between the iterated (anti)derivatives of e s·x 1/2 with respect to x and spherical modified Bessel functions of second kind

Abstract

Abstract This paper investigates the underlying structure of the n-th derivative of e s·x 1/2 with respect to x. For n ∈ ℤ+, (dn /d xn )e s·x 1/2 can be expressed in terms of spherical modified Bessel functions of second kind in the complex plane. The representation holds for n ∈ ℤ-, where it represents the particular n-th antiderivative of e s·x 1/2 with all n constants of integration equal to zero. Our results introduce new connections among mathematical applications and provide some Bessel properties.