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## Homework Statement

I am working through some maths to deepen my understanding of a topic we have learnt about. However I am not sure what the author has done and I have copied below the chunk I am stuck on. I would be extremely grateful if someone could just briefly explain what is going on i.e how to get from one step to another and why.

Thanks

## Homework Equations

$\int^\infty_y P(2y;2\nu,2k) dK$

where $P(2y;2\nu,2k)$ =$ \frac{1}{2} (\frac{x}{\lambda})^{\frac{\nu-2}{4}} I_{\frac{\nu-2}{2}}(\lambda x)^{\frac{1}{2}} e^{\frac{-(\lambda+x)}{2}}$

Note that I is the bessel function of the first kind of order K and is defined as $ (\frac{1}{2}Z) \sum\limits_{j=0}^\infty \frac{(\frac{z^2}{4})^j}{{j!\gamma(k+j+1)}}$

$$=\int^\infty_y e^{-z-k} (\frac{z}{k})^{\tau-1} (kz)^{\frac{\tau-1}{2}} \sum\limits_{n=0}^\infty \frac{(zk)^n}{n!\gamma(n+\nu-1+1)}dK$$

$$=\int^\infty_y \frac{e^{-z} z^{n+v-1}}{\gamma(n+\nu)} \int^\infty_y \frac{e^{-k} k^{n}}{\gamma(n+1)}dK$$

$$=\sum\limits_{n=0}^\infty g(n+v,z)G(n+1,y)$$

Please also note that $\gamma$ is meant to be that symbol that looks like a T when working with gamma funtions but I do not know what it is.

## The Attempt at a Solution

**The only thing I know is that it has something to do with gamma functions and integration by parts.**

By the way this is not assignment related but something I really want to understand.

By the way this is not assignment related but something I really want to understand.