The integral of the product of a power and Bessel’s $K_{\nu}$ function

Abstract

Withers and Nadarajah [Braz. J. Probab. Stat. 28 (2014) 140–149] gave new expressions for hypergeometric functions when two arguments differ by an integer. Here, we give new expressions for $\int^{x}_{x_{0}}x^{\mu}K_{\nu}(x)\,dx$ when $\mu \pm \nu$ is an integer, where $K_{\nu}(\cdot)$ denotes the modified Bessel function of order $\nu$. Each new expression is a finite sum of terms involving only the gamma function and the modified Bessel function.