Repeated Integrals and Derivatives of K Bessel Functions

Repeated integrals of Bessel functions $K_\nu (x)$ on $0 < x < \infty $, denoted by $Ki_{\nu ,n} (x)$, are considered. Series are derived in terms of exponential integrals that are direct extensions of known results for the Bickley functions $Ki_n (x)$. The basic result for $n = 0$ can also be extended to $n < 0$ by repeated differentiation. For $\nu $ a nonnegative integer, it is also shown that $Ki_{\nu ,n} (x)$ can be represented as a finite sum of Bickley functions of which $Ki_{\nu ,0} (x) = K_\nu (x)$ is a special case.