Definite intergal with two K-Bessel functions and x I would like to calculate the definite integral with K-Bessel funcitons and a and b complex (n and k integers): 
$$\int_{0}^{\infty} x \;K_{a}(nx) \; K_{b}(kx) \; dx$$
I could not find it in litterature with a and b complex (We have $Re(a)<1$ and $Re(b)<1$ for convergence in zero!).
Any reference or help on this subject is welcome. 

Mathematica:
$$\int_{0}^{\infty} x \;K_{a}(nx) \; K_{b}(kx) \; dx=\frac{1}{2n^2}(k n)^{-b} $$
$$\qquad\times \left[n^{2 b} \Gamma (b) \Gamma \left(-\frac{a}{2}-\frac{b}{2}+1\right) \Gamma \left(\tfrac{1}{2} (a-b+2)\right) \, _2F_1\left(\tfrac{1}{2} (-a-b+2),\tfrac{1}{2} (a-b+2);1-b;\frac{k^2}{n^2}\right)\right.$$
$$\qquad\left.+\,k^{2 b} \Gamma (-b) \Gamma \left(\tfrac{1}{2} (-a+b+2)\right) \Gamma \left(\tfrac{1}{2} (a+b+2)\right) \, _2F_1\left(\tfrac{1}{2} (-a+b+2),\tfrac{1}{2} (a+b+2);b+1;\frac{k^2}{n^2}\right)\right]$$
for $-2<{\rm Re}(a-b)<2$, $-2<{\rm Re}(a+b)<2$
if $k=n$ this simplifies to
$$\int_{0}^{\infty} x \;K_{a}(nx) \; K_{b}(nx)=\frac{\pi ^2 (a^2-b^2)}{4 n^2 (\cos \pi  b-\cos \pi  a)}$$
there do not seem to be further simplifications if $k\neq n$
