Does anyone have an idea to compute the following integral? $$ I=\int_0^\infty\frac{z^{m+n+1} K_{n-m}\left(a z\right)}{z^2+1}dz $$ where $m\geq n$ are positive integers, and $a>0$.
Since $m,n$ are integers and $n-m\leq 0$ (which is in the Bessel function) is non-positive, I could not find any matching solution in integral books.