Hi I am looking to find
$$\int_{0}^{\infty} \lambda^{k+p-1}e^{-(a\lambda+b/\lambda)/2 - \lambda} d\lambda$$
where $k$ is an integer.
I'm stuck, I've even tried to relax $p$ to an integer too so I could apply Ramanujan's Master Theorem however there is no Taylor series for $e^{-(a\lambda+b/\lambda)/2 - \lambda}$ at $0$ which makes it even harder to solve for the simpler case as I have no clue how else I could find $\phi$.
Please could you help or give me a hint in the right direction?
Thanks
If you look at formula $8.432.6$ in the "Table of Integrals, Series and Products" (seventh edition) by I.S. Gradshteyn and I.M. Ryzhik.
$$K_{\nu }(z)=\frac{1}{2}\left(\frac{z}{2}\right)^{\nu }\int_0^\infty t^{-(\nu+1)}\, e^{-t-\frac{z^2}{4\, t}}\,dt$$ $$\frac{1}{2} \left(a \lambda +\frac{b}{\lambda }\right)+\lambda=\left(1+\frac{a}{2}\right) \lambda +\frac{b}{2 \lambda }$$ So, make $$\left(1+\frac{a}{2}\right) \lambda=t\implies \lambda=\frac{2 t}{a+2}$$ Then, your integral is $$\left(\frac{2}{a+2}\right)^{k+p}\int_0^\infty t^{k+p-1}\,\,e^{-t-\frac{(a+2) b}{4 \,t}}\,dt$$ Just define now $z$ and you are done.