I am looking to derive the non-central chi square density function. For this I have the moment generator function: $$m.g.f= (1-2t)^{-n/2} \exp[\frac{\lambda t}{1-2t}]$$ and I know that the p.d.f of $X^2$ (where $X\sim\mathcal{N}(\mu,1)$ is: $$f_{X^2}(x) = \frac{e^{-(x+\mu^2)/2} \cosh (\mu \sqrt{x})}{\sqrt{2\pi x}}, \quad x \ge 0.$$
Wikipedia says that I can expand the cosh term using Taylor series to get this distribution function, but it is not clear to me how I should do it.